Math 317: Linear Algebra Notes and Problemsfaculty.sfasu.edu/longne/index_files/LAIBL/LA...

Math 317: Linear AlgebraNotes and Problems

Nicholas Long

SFASU

Contents

Introduction iv

1 Efficiently Solving Systems of Linear Equations and Matrix Op-erations 11.1 Warmup Problems . . . . . . . . . . . . . . . . . . . . . . . 11.2 Solving Linear Systems . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Geometric Interpretation of a Solution Set . . . . . . 91.3 Vector and Matrix Equations . . . . . . . . . . . . . . . . . 111.4 Solution Sets of Linear Systems . . . . . . . . . . . . . . . 151.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 181.6 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . 20

1.6.1 Special Types of Matrices . . . . . . . . . . . . . . 211.6.2 Matrix Multiplication . . . . . . . . . . . . . . . . . 22

2 Vector Spaces 252.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . 312.4 Linear Transformations . . . . . . . . . . . . . . . . . . . . 332.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Connecting Ideas 403.1 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 rank and nullity . . . . . . . . . . . . . . . . . . . . 413.1.2 Coordinate Vectors relative to a basis . . . . . . . . 42

3.2 Invertible Matrices . . . . . . . . . . . . . . . . . . . . . . 443.2.1 Elementary Matrices . . . . . . . . . . . . . . . . . 44

ii

CONTENTS iii

3.2.2 Computing Inverses . . . . . . . . . . . . . . . . . 453.2.3 Invertible Matrix Theorem . . . . . . . . . . . . . . 46

3.3 Invertible Linear Transformations . . . . . . . . . . . . . . 473.4 LU factorization of matrices . . . . . . . . . . . . . . . . . 483.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.1 Computing Determinants . . . . . . . . . . . . . . . 493.5.2 Properties of Determinants . . . . . . . . . . . . . . 51

3.6 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . 523.6.1 Diagonalizability . . . . . . . . . . . . . . . . . . . 553.6.2 Eigenvalues and Eigenvectors of Linear Transfor-

mations . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Inner Product Spaces 584.1 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Orthogonal Complements . . . . . . . . . . . . . . . . . . . 594.3 QR Decompositions . . . . . . . . . . . . . . . . . . . . . . 59

Nicholas Long

Introduction

In this course you will learn about linear algebra by solving a carefullydesigned sequence of problems. It is important that you understand everyproblem and proof. As hard as it is to imagine, you will occasionally wantto have more questions to work on in order to fully understand the ideas inthis class. Feel free to ask me about additional problems to consider whenyou are stuck. The notes and problems will refer back to previous parts, so Isuggest you keep a binder with the notes and your work together and bringall of these materials to class and any office hours. I also suggest havingseveral colored pens or pencils to help you draw and label your work so itwill be easily understood. Your written work needs to be legible and largeenough so that someone else can easily read and understand what you aredoing.

Unlike mathematics courses you have had in the past, solving a problemin this course will always have two components:

• Find a solution

• Explain how you know that your solution is correct.

This will help prepare you to use mathematics in the future, when there willnot be someone to tell you if your solution is correct or not. That is alsowhy it will be primarily up to you (the students) to assess the correctnessof work done and presented in class. This means that using the proper lan-guage (both written and spoken) and specificity are keys to effective com-munication. This class will teach you about the specificity and precisionof mathematical language, so it is important that you practice and work onthis. In order for you to understand the ideas in this class you will need toevaluate other people’s ideas as well as your own for correctness. The workin this class is not about getting an answer but rather making sense of theprocess and ideas involved in the problems. For this reason, justification inyour work and ideas is very important. Why you did or tried something isjust as important as what you did or what result you got. In fact, clearlyarticulating your thought process will make you a more efficient thinker.

iv

Introduction v

You are not alone in this course. The role of the instructor is to guide thediscussion and make sure you have the resources to be successful. Whilethis new learning environment can be a bit unsettling for students at first,you will get comfortable as you do more problems and get feedback fromother students and the instructor. I am also here to help you outside of classtime and expect you to find a way to get the help you need, whether faceto face or over email. You will find that once you have thought about aproblem for a little while, it will only take a little push to get unstuck.

Some Notation:

• N= {0,1,2,3, ...} is the set of natural numbers.

• Z= {...−3,−2,−1,0,1,2,3, ...} is the set of integers.

• R is the set of real numbers.

• Rn is the set of vectors with n components from R. Rn can also beviewed as the set of points in n-space. For instance, R2 would be thecartesian plane and R3 would correspond to three dimensional space.

• C= {a+bı|a,b ∈ R} is the set of complex numbers.

• A, B, C, ... denote matrices.

• ~u, ~v, ~w, ... denote vectors (which starting in Chapter 2 will not neces-sarily be in Rn).

• ~0V is the zero vector in vector space V .

• /0 is the null set or set that contains no elements.

• Mm×n is the set of matrices with m rows and n columns.

• DNMS is an acronym for ’Does not make sense’.

• DNE is an acronym for ’Does not exist’.

• #pf describes productive failure

Definitions will be bolded for some terms and others will have their ownheading and number. Many definitions and comments will be numbered sothat everyone will be able to refer to them in work.

You will need to make some plots in this class and it will be advanta-geous to use a computer algebra system in some problems. For this rea-son, I suggest that if you have not already, you should make an account onthe Sage MathCloud at https://cloud.sagemath.com. Additionally, the SageMath Cloud will allow you to easily write your work in LaTex, a wonderfultypsetting program. A homework bonus will be given to students who writetheir work in LaTex.

Nicholas Long

Chapter 1

Efficiently Solving Systems of Linear Equationsand Matrix Operations

1.1 Warmup Problems

Question 1. Solve:3x1−2x2 = 6

−x1 + x2 = 1

Question 2. Draw a graph of the solution set of the equation: 3x1−2x2 = 6(Hint: If a solution has x1 = a, what is x2 or viceversa?)

Question 3. Draw a graph of the solution set of the equation: −x1+x2 = 1

Question 4. Graph the solution sets from Questions 2 and 3. How doesyour answer to Question 1 compare to your graph?

Question 5. Solve:2x1−2x2 = 6

−x1 + x2 = 1

Question 6. Solve:2x1−2x2 =−2

−x1 + x2 = 1

Question 7. Wait... what just happened? Explain the results of the previoustwo problems. What do the graphs of the corresponding solution sets looklike in relation to the graphs of the equations?

Question 8. What are the possible intersections of two lines? Clearly stateyour conjecture.

1

Efficiently Solving Systems of Linear Equations and Matrix Operations 2

Throughout this course we will be doing many of the same things youdid in the previous questions, but we will do them in a more general settingthat will allow us to solve many new and old problems.

Nicholas Long


1.2 Solving Linear Systems

Our first chapter will cover the ideas of efficiently solving a system of linearequations and matrix operations.

A system of m linear equations in n variables can be written:

a11x1 + a12x2 + ... + a1nxn = b1a21x1 + a22x2 + ... + a2nxn = b2

......

...am1x1 + am2x2 + ... + amnxn = bm

The term ai j is the coefficient of the j-th variable (denoted x j) in the i-thequation. A solution is choice of variable values that satisfy all equationsin the system. A solution is not a particular variable value but must includea choice for all variables in the system. The solution set for a system ofequations is the set of all possible solutions. We will have many ways todescribe solutions to a system this semester but they all specify the valuesof x1, x2, ..., and xn, sometimes as an ordered n-tuple (x1, x2, ..., xn).

Question 9. Is (1,2,3) a solution to the following system?

1x1 + 2x2 + 3x3 = 142x1 − 3x2 + 2x3 = 0x1 + 7x3 = 0

The previous problem shows how easy it is to check if a set of variablevalues is a solution, but finding a solution or the set of all solutions is harderbut very important to many problems. Generally speaking, the process offinding the solution set for a system of equations is to trade the system ofequations you have for an equivalent system (a system with the same solu-tion set). Hopefully it will be easier to explicitly write the solution set of thenew equivalent system. An elementary operation on a system of equationsis an operation of the form:

a) multiplying an equation by a non-zero scalar

b) switching two equations

c) adding a multiple of one equation to another equation

Question 10. Given a set of equations in three variables show that elemen-tary operations give an equivalent system of equations. In other words, youneed to show that (c1,c2,c3) is a solution of the equations

a1x1 +a2x2 +a3x3 = a4

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b1x1 +b2x2 +b3x3 = b4

if and only if (c1,c2,c3) is a solution to the equations obtained after per-forming one of the operations above. Hint: Try part c) first.

Remark 11. For those of you new to the term “if and only if”(sometimesabbreviated iff), if and only if denotes a biconditional statement. A state-ment of the form “P iff Q”implies “if P, then Q”and “if Q, then P”. Ifyou are asked to prove a biconditional statement you need to prove bothconditional statements. The good news is that if you have a biconditionalstatement as a theorem, then you can use either (or both) of the conditionalstatements in your work.

Question 12. Solve the following system just using elementary operations.Remember to show your work.

3x−2y− z = 0

2x+ y+ z = 10

x+4y+3z = 20

A system of equations is consistent if there exists at least one solution tothe system. In other words, a consistent system of equations has a nonemptysolution set. A system that is not consistent is said to be inconsistent.

In the Question 12, note that you didn’t change anything but the coef-ficients in the system of equations as you traded one system for another.Some of the coefficients probably became 0 but you didn’t really eliminateany variables or consider a totally different problem. We will use matri-ces to efficiently store, and manipulate the coefficients in a system of linearequations, since they are all that matter for now. Matrices will have manyuses in this and other courses, and we will use capital letters like A and B todenote matrices. Matrices will be rectangular arrays with the same numberof entries in each row and the same number of entries in each column. Thesize of a matrix is given (in order) as the number of rows by the number ofcolumns, so a 3 by 2 matrix has 3 rows and 2 columns.

In order to specify what entry we are referring to in a matrix, we need anordered pair of indices telling us the number of the row and number of the

column to look in respectively. For instance, if B =

1 5 0♥ F �£ r z

, then the

(3,2) entry of B is in the third row and 2nd column. You could also writethis as (B)3,2 =r. The i-th row of a matrix A will be denoted rowi(A) andthe j-th column will be denoted column j(A).

Nicholas Long


In order to distinguish vectors (as being more than just n by 1 matrices),we will use the arrow notation and lower case symbols like~u and~v to denotevectors. Unless otherwise stated, we will use column vectors. For instance,

if ~v =

v1v2...

vm

, then the second component of ~v is the scalar v2. The size of

a vector in Rn is the number of components the vector has. In Chapter 2,we will deal with a much more general notion of vectors that will not havecomponents like vectors in Rn.

The coefficient matrix of a linear system of m equations in n variables isa m by n matrix whose (i, j) entry is the coefficient of the j-th variable, x j,in the i-th equation of the system. The augmented matrix of a linear systemof m equations in n variables is a m by (n+1) matrix whose first n columnsare the coefficient matrix of the system and the last column is the constantterms from the right side of each equation.

The system


......

......

am1x1 + am2x2 + ... + amnxn = bm

has a coefficient matrix

A =

a11 a12 ... a1na21 a22 ... a2n

......

...am1 am2 ... amn

and an augmented matrix of

[A|b] =

a11 a12 ... a1n | b1a21 a22 ... a2n | b2

......

... | ...am1 am2 ... amn | bm

For some properties of the system of equations, we need only look at the

coefficient matrix but others will need the augmented matrix. It is importantto know the difference and be able to state which corresponding matrix youare using in your work.

Nicholas Long


Question 13. What is the coefficient matrix for the previous systems? Givethe coefficient matrix for Question i as Ai. (Hint: You should have 8 answersto this question.)

Question 14. What is the augmented matrix for the previous systems? Givethe augmented matrix for Question i as Ai. (Hint: You should have 8 an-swers to this question.)

The elementary operations on equations outlined above will correspondto elementary row operations on matrices as well. Specifically, an elemen-tary row operation on a matrix is an operation of the form:

• multiplying a row by a non-zero scalar

• switching two rows

• adding a multiple of one row to another row

We now have operations to trade our system of equations for an equivalentsystem, but we have not stated a way to make sure that the solution set willbe easy to explicitly state from our new equivalent system. The followingmatrix forms will be useful for determining solution sets and various otherproperties of the corresponding system of equations.

A rectangular matrix is in row echelon form if it has the following threeproperties:

a) All nonzero rows are above any rows of all zeros.

b) Each leading entry of a row is in a column to the right of the leadingentry of the row above it.

c) All entries in a column below a leading entry are zeros.If a matrix in row echelon form satisfies the following additional prop-erties, then we say the matrix is in reduced row echelon form:

d) The leading entry in each nonzero row is 1.

e) Each leading 1 is the only nonzero entry in its column.

The leading entry in a nonzero row of the row echelon form is called apivot. The column in which a pivot occurs is called a pivot column andthe corresponding variable is a basic variable or pivot variable. A variablecorresponding to a column in which the coefficient or augmented matrixdoes not have a pivot are called free variables. While the echelon formis needed to find where pivots will occur, we will sometimes refer to pivotpositions of a matrix even when the matrix is not in echelon form.

Nicholas Long


Theorem 15. The reduced row echelon form of a rectangular matrix isunique.

It is important to note that the row echelon form of a matrix is not unique.

Question 16. Give an example of a matrix M that has the following prop-erties. If such a matrix cannot exist, explain why.

a) M satisfies properties a and b of row echelon form but does not satisfyproperty c.

b) M satisfies properties a and c of row echelon form but does not satisfyproperty b.

c) M satisfies properties b and c of row echelon form but does not satisfyproperty a.

d) M satisfies properties a, b, and c of row echelon form but does notsatisfy property d of reduced row echelon form.

e) M satisfies properties a, b, c, and d of reduced row echelon form butdoes not satisfy property e of reduced row echelon form.

Question 17. List out all possible row echelon forms of 3 by 4 matricesusing the symbols � for a pivot, ∗ for a non-pivot entry (possibly 0), and 0(when an entry must be 0). Hint: There are 15 possible.

Question 18. List out all possible reduced row echelon forms of 3 by 4matrices using the symbols � for a pivot, ∗ for a non-pivot entry (possibly0), and 0 (when an entry must be 0). What value must the � entries be?

Question 19. Solve each of the following systems by converting to an aug-mented matrix and using elementary row operations to reduce the aug-mented matrix to reduced row echelon form. With each reduced row ech-elon form, put a box around all pivot entries. Use the system of equationscorresponding to the reduced row echelon form to write out the solution setfor each system.

a)3x1−2x2 = 6

2x1−2x2 =−2−x1 + x2 = 1

b)3x1−2x2 = 62x1−2x2 = 6−x1 + x2 = 1

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c)4x− y+3z = 5

3x− y+2z = 7

d)7x−11y−2z = 3

8x−2y+3z = 1

e)3r−5s+ t = 2

−6r+10s−2t = 3

Question 20. Given an augmented matrix [A|b] and its row reduced echelonform [A′|b′], write a procedure (set of steps) to give the solution set to thecorresponding systems of equations. Make sure you are clear in writing outyour steps.

Are there any ways in which your procedure will fail to work? Are thereany systems with augmented matrices which will not have a reduced rowechelon form?

Two of the most important questions we will consider this semester are:1) Is the system consistent?2) If a solution exists, is the solution unique?

Question 21. Look back at your results so far and try to figure out whatproperties of the system (or corresponding matrices) will help us answerquestion 1 and which properties of the system will help us answer question2. Write a conjecture about each question.

The space below is for you to write a statement of the theorem that theclass decides is the best for determining consistency and uniqueness.

Theorem 22. Classwide statement of consistency theorem:

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Theorem 23. Classwide statement of uniqueness theorem:

The following two proofs are difficult and one of the reasons we will beresubmitting proofs.

Question 24. Prove the classwide statement of the consistency theorem.

Question 25. Prove the classwide statement of the uniqueness theorem.

Question 26. Using the statement of the classwide uniqueness and consis-tency theorems, treat each of your answers to Question 17 as a coefficientmatrix of a linear system of equations and state:

a) whether each corresponding system of equations will be consistent,inconsistent, or you can’t tell.

b) whether each corresponding system of equations will have a uniquesolution, multiple solutions, no solutions, or you can’t tell.

Question 27. Using the statement of the classwide uniqueness and consis-tency theorems, treat each of your answers to Question 17 as an augmentedmatrix of a linear system of equations and state:

a) whether each corresponding system of equations will be consistent,inconsistent, or you can’t tell.

b) whether each corresponding system of equations will have a uniquesolution, multiple solutions, no solutions, or you can’t tell.

1.2.1 Geometric Interpretation of a Solution Set

Recall from Question 2-8, that the solution set of a linear equation in twovariables was a line in R2 (the plane) and that the solution set of a system

Nicholas Long


of two equations in two variables was possibly a point, a line, or empty.Similarly, the solution set for a linear equation in three variables will be aplane in 3-space (R3).

Question 28. List out all the possible ways m planes can intersect in athree dimensional space. This answer may change depending on m, so becomplete in your answer.

We don’t usually draw what a solution set of a linear equation in fourvariables looks like because drawing in 4D is difficult. The graph wouldbe called a hyperplane in 4-space. For the same reasons, we certainly don’tdraw m hyperplanes in n-space but the solution sets work very similar to thepictures we can draw in 3-space (also known as R3).

Question 29. Why does the graph of a linear equation in n variables needto be a flat n−1 dimensional hyperplane?

Recall that the following commands will plot the plane given by 3x−2y− z = 0 in Sage:

var(’x,y’);

plot3d(3*x-2*y,(x,-2,2),(y,-2,2),color=’red’)

Plotting the equations, 3x−2y−z = 0, 2x+y+z = 10, and x+4y+3z =20 in red, yellow, and green respectively gives:

Question 30. Does your answer to Question 12 make sense with this plot?

Nicholas Long


Question 31. For each of the systems in Question 19, use Sage to draw aplot of each of the equations in the system and write a sentence for eachsystem about why the plot and your answer to Question 19 make sense.

If you remember parametric equations of lines and planes in space frommultivariable calculus, then we will return to those ideas in Section 1.4.

1.3 Vector and Matrix Equations

Recall that two vectors in Rn are equal if and only if all of their componentsare equal.

Question 32. Prove that the system of equations given by


......

......


has the same set of solutions as the vector equation

x1

a11a21

...am1

+ x2

a12a22

...am2

+ · · ·xn

a1na2n

...amn

=

b1b2...

bm

In other words, show that (c1,c2, ...,cn) is a solution to the vector equationiff (c1,c2, ...,cn) is a solution to the system of linear equations. Make sureyou clearly connect the ideas in your proof and do not make an argumentthat these equations look similar.

Question 33. Solve the following vector equation directly:

x1

[11

]+ x2

[−11

]=

[3−4

]

Question 34. Give an example of a vector

abc

such that the equation

x1

110

+ x2

001

=

abc

has no solution, or explain why no such vector exists.

Nicholas Long



abc


x1

110

+ x2

001

=

abc

has exactly 1 solution, or explain why no such vector exists.


abc


x1

110

+ x2

001

+ x3

111

=

abc



abc


x1

110

+ x2

001

+ x3

111

=

abc

has no solutions, or explain why no such vector exists.


abc


x1

110

+ x2

001

+ x3

abc

=

70−2


A linear combination of a set S is a vector of the formn

∑i=1

ci~vi = c1~v1 + c2~v2 + ...+ ck~vk

where~vi ∈ S. Note thatn

∑i=1

ci~vi

will not usually be in S even though~vi ∈ S.

Nicholas Long


Question 39. Can you write b =

[24

]as a linear combination of v1 =

[11

]and v2 =

[−11

]? Justify your answer.

Question 40. Repeat the previous problem for b =

[00

]and b =

[3−1

].

Question 41. Can you write b =

204

as a linear combination of v1 =

111

and v2 =

−111

? Justify your answer.

Definition 42. We define a matrix-vector product as follows:If A is a m by n matrix,

A =

a11 a12 ... a1na21 a22 ... a2n

......

...am1 am2 ... amn

and~x =

x1x2...

xn

∈ Rn, then the matrix-vector product

A~x = x1

a11a21

...am1

+ x2

a12a22

...am2

+ · · ·xn

a1na2n

...amn

Question 43. If A is a m by n matrix, then A~x ∈ R♦ for what ♦?

Based on the above definition of the matrix vector product, if

A =

a11 a12 ... a1na21 a22 ... a2n

......

...am1 am2 ... amn

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and~b =

b1b2...

bm

, then by Question 32, A~x =~b has the same solution set as the

systema11x1 + a12x2 + ... + a1nxn = b1a21x1 + a22x2 + ... + a2nxn = b2

......

......


Question 44. a) Write out the k-th component of the resulting vector ofthe product

a11 a12 ... a1na21 a22 ... a2n

......

...am1 am2 ... amn

x1x2...

xn

b) How can you express the result of the matrix-vector product in terms

of~x and the rows of A?

c) How can you express the result of the matrix-vector product in termsof~x and the column of A?

Question 45. Write each of the following as a matrix equation, a vectorequation, and system of equations. You need to write out the exact cor-responding vector equation, matrix equation, and system of equations, notsome equivalent form.

a)

1 2 34 5 67 8 9

x1x2x3

=

abc

b) a1

125

+ 3

0−1

+a2

−102

=

450

c)

2x1 +3x2 = 7

−x1 + x2 +4x3 = 0

5x1−6x2− x3 = 2

Nicholas Long


1.4 Solution Sets of Linear Systems

In this section, we will talk about efficient and clear ways to express the setof solutions to a linear system of equations.

Question 46. Prove that if a system of linear equations has two distinctsolutions, then the system has infinitely many solutions.

Question 47. For each of the systems in Question 19, use the reduced rowechelon form to solve for each pivot (basic) variable in terms of the freevariables and constant terms. By substituting in your new expresssions for

the pivot variables into the vector

x1x2...

xn

, you will get a vector solely in

terms of the free variables. You can now write the solution set as a linearcombination of vectors with each free variable as a coefficent. For instance,if a system had free variables x j and xk, then the parametric form wouldlook like ~u+ x j~v1 + xk~v2. This is called the parametric form of the solutionset for the system.

Question 48. Solve each of the following systems and present your solutionset in parametric form.

a)3x1−2x2 = 0

2x1−2x2 = 0

−x1 + x2 = 0

b)3x1−2x2 = 0

2x1−2x2 = 0

−x1 + x2 = 0

c)4x− y+3z = 0

3x− y+2z = 0

d)7x−11y−2z = 0

8x−2y+3z = 0

Nicholas Long


e)3r−5s+ t = 0

−6r+10s−2t = 0

Definition 49. A system of linear equations is homogeneous if the matrix

form of the system, A~x=~b has~b=~0=

0...0

. If~b 6=~0, then the corresponding

system is nonhomogeneous.

Question 50. Using your answers to Questions 47 and 48 as a guide, stateand prove a theorem that discusses how the solution set to a homogeneoussystem is related to the solution set of the non-homogenous system.

Question 51. State and prove a theorem that describes under what con-ditions of the matrix A the system A~x =~b will have a solution for every~b. Essentially, you need to fill in the blank of the following statement: If

, then A~x =~b will have a solution forevery~b ∈ Rm.

Definition 52. The column space of a matrix A, denoted Col(A) is the setof vectors that can be written as a linear combination of the columns of A.If A is m by n, then Col(A) = {~b ∈ Rm|A~x =~b for some~x ∈ Rn}.

Theorem 53. The pivot columns of a matrix A generate Col(A). This meansthat if~v ∈Col(A), then~v can be written as a linear combination of the pivotcolumns of A.

Note that this theorem uses the pivot columns of A and not the pivotcolumns of the echelon form of A.

Definition 54. The null space of a matrix, denoted Null(A), is the set ofvectors that when multiplied by the matrix give the zero vector. In otherwords, Null(A) is the solution set of the homogeneous equation A~x =~0.

Question 55. Let A =

[1 2 34 5 6

]. Describe the sets Col(A) and Null(A)

using a parametric form.


1 23 45 6

. Describe the sets Col(A) and Null(A) using

a parametric form.

Nicholas Long


Question 57. Let

A =

−3 6 −1 1 −7 01 −2 2 3 −1 02 −4 5 8 −8 0

∼

1 −2 0 −1 0 00 0 1 2 0 00 0 0 0 1 0

What is the reduced row echelon form of

−3 −1 11 2 32 5 8

? You should use

the information given above and not a lot of calculations.

Question 58. Let

A =

−3 6 −1 1 −71 −2 2 3 −12 −4 5 8 −8

∼

1 −2 0 −1 00 0 1 2 00 0 0 0 1

Describe Col(A) and Null(A) using a parametric form using as few vectorsas possible.

Question 59. Under what conditions on A, a m by n matrix, will Col(A) beall of Rm?

Remember that Col(A) and Nul(A) are usually very different sets, in fact,they aren’t always in the same parent set. If A is a m by n matrix, then Col(Ais in R and Nul(A) is in R .

Question 60. Find an example of a 2 by 2 matrix where Col(A) is the sameset as Nul(A).

Question 61. Given a matrix A with echelon form

� ∗ ∗ ∗0 0 � ∗0 0 0 0

:

a) What is the minimum number of vectors that will be needed to give theparametric form of Col(A)?

b) What is the minimum number of vectors that will be needed to give theparametric form of Null(A)?

c) Col(A) is a subset of R♠ for what value of ♠?

d) Null(A) is a subset of R♠ for what value of ♠?

Question 62. Given a matrix A with echelon form

0 � ∗ ∗ ∗0 0 0 � ∗0 0 0 0 0

:

Nicholas Long


a) What is the minimum number of vectors that will be needed to give theparametric form of Col(A)?

b) What is the minimum number of vectors that will be needed to give theparametric form of Null(A)?

c) Col(A) is a subset of R♠ for what value of ♠?

d) Null(A) is a subset of R♠ for what value of ♠?

Question 63. Given a matrix A, how many vectors will be needed to givethe parametric form of

• Col(A)

• Null(A)

Question 64. Can every vector in R3 be written as a linear combination of

the columns of A =

1 5 −2 0−3 1 9 −54 8 −1 7

? Justify your answer.

1.5 Applications

Solving linear systems of equations can be used to balance equations ofchemical reactions. For instance, if propane (C3H8) and oxygen (O2) reactto give off carbon dioxide (CO2) and water (H2O), you can write somethinglike C3H8 +O2 → CO2 +H2O but this is not balanced because there arethree carbon atoms going into this reaction and only one coming out. Insteadwe look at x1(C3H8)+ x2(O2)→ x3(CO2)+ x4(H2O), where xi stands forthe number of molecules of each reactant needed to balance the chemicalequation. We can then solve a system of equations where each equationcomes from balancing a different element involved in the reaction.

Question 65. Give the system of equations described by the burning ofpropane above. Solve the system and check that this balances the chemi-cal equation.

Question 66. Balance the following equations:

a) B2S3 +H2O→ H3BO3 +H2S

b) KMnO4 +MnSO4 +H2O→MnO2 +K2SO4 +H2SO4

Question 67. Should the systems used to balance a chemical equation havea unique solution? Why or why not? Consider how many equations you willhave and how many variables.

Nicholas Long


Question 68. In studying the urban-rural migration patterns of Springfield,you notice that in any given year 70% of the population in urban areasstays in urban areas, 20% moves to suburban areas, and 10% moves torural areas. For population in a suburban area, the respective percentagesmoving to urban, suburban, and rural areas are 5%, 80%, and 15%. Forpopulations in a rural area, the respective percentages are 11%, 14%, and75%.

a) If ut , st , and rt are the percentages of the population in an urban,suburban, and rural areas in year t, set up a system of linear equationsfor ut+1, st+1, and rt+1 in terms of ut , st , and rt .

b) Write your answer to part a) as a matrix equation where ~xt =

utstrt

and ~xt+1 =

ut+1st+1rt+1

.

c) Use Sage to compute ~x1,~x2, ..., ~x10, when ~x0 =

.4.2.4

.

Question 69. Networks (traffic, electrical, etc.) can be studied by settingup a system of linear equations corresponding to the different junctions.Specifically, at each junction, the amount of stuff coming in (flow in) mustbe the same as the amount of stuff going out (flow out). If x1, ...,x9 arethe amount of stuff flowing between junctions, set up a system of equationscorresponding to the network given below.

Question 70. Is there only one way that the flow of stuff can satisfy theinformation given by the network graph above? Why or why not?

Nicholas Long


Question 71. Bonus: Solve the system of equation that is your answer toQuestion 69. Describe how the parts of your solution set correspond to thenetwork.

1.6 Matrix operations

Just as you can add two vectors in Rn componentwise, you can add twomatrices entry-wise. For this reason, it only makes sense to add two matricesif they are the same size. You can also define scalar multiplication of amatrix entry-wise.


3 −1 02 0 −74 2 6

, B =

6 −2 03 0 −214 2 6

,

and C =

1 0 00 1 01 1 1

.

a) Is B a scalar multiple of A? Why or why not?

b) 2A−3C =

c) −(A+C)+2B =

d) (84A+16B−12C)2,1 =

Question 73. Symbolically, (A+B)i, j = and (kA)i, j =

Definition 74. Let A be a m by n matrix. The transpose of A, denoted AT , isa n by m matrix such that (AT )i j = (A) ji.

There are a couple of ways to think about the transpose. First, you canthink about flipping the matrix across the main diagonal (the elements ofthe form Ai,i). You can also view the transpose of a matrix as switching therows and columns (but preserving the order). In other words, the i-th row ofAT is the i-th column of A.


3 −12 04 2

and B =

[6 −2 03 0 −21

]

a) AT =

b) BT

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c) AT +B

d) BT +A


3 −1 02 0 −74 2 6

, B =

6 −2 03 0 −214 2 6

,

and C =

1 0 00 1 01 1 1

.

a) A+BT =

b) ((C−B)T +A)T =

Question 77. Prove that (A+B)T = AT +BT where A and B are m by nmatrices.

Question 78. When does it make sense to add A and AT ?

Question 79. Prove that (AT )T = A.

A matrix A is symmetric if AT = A.

Question 80. Prove that the sum of two symmetric m by n matrices is sym-metric.

Question 81. Prove that kA is symmetric if A is symmetric.

1.6.1 Special Types of Matrices

A m by n matrix A is said to be upper triangular if (A)i j = 0 wheneveri > j. Similarly, a matrix A is lower triangular if (A)i j = 0 whenever i < j.

Question 82. Give an example of a matrix that is upper triangular but isnot in echelon form. If one does not exist, explain why.

Question 83. Give an example of a matrix that is in echelon form but is notupper triangular. If one does not exist, explain why.

Question 84. Can a matrix be upper and lower triangular? Either give anexample or explain why there cannot exist one.

Diagonal matrices are matrices whose only nonzero elements are on thediagonal. Specifically, a matrix A is diagonal if (A)i j = 0 whenever i 6= j.

Question 85. Give an example of a matrix that is diagonal but not in eche-lon form.

Nicholas Long


The n by n identity matrix, denoted Idn, is the unique matrix such thatIdn~x =~x for every ~x ∈ Rn. In fact the entries of Idn are easily computed interms of the Dirac delta function. Specifically (Idn)i, j = δi, j, where

δi, j =

{0 if i 6= j1 if i = j

Question 86. Write out Id5 and use it to prove that for any ~v ∈ R5 theproduct of Id5 and~v will always be~v.

Question 87. Bonus: Prove that Id5 is the only matrix that has the propertyfrom the problem above.

1.6.2 Matrix Multiplication

Earlier we saw how to multiply a m by n matrix by a vector from Rn. We willdiscuss how to define matrix multiplication with multiple interpretations.

Let A be an m by n matrix and let ~x1 and ~x2 be vectors from Rn. Earlierwe defined what A~x1 and A~v2 meant. If we build a n by 2 matrix B using ~x1and ~x2 as the columns, then we can define AB, read as ”A times B”, to be

AB = A[~x1 ~x2] = [A~x1 A~x2]

The above definition is just distributing our matrix-vector product across thecolumns of B. In a similar fashion, given any n by k matrix

B = [~b1 ~b2 · · · ~bk]

where ~bi is the i-th column of B, we can define

AB = [A~b1 A~b2 · · · A~bk]

In particular, this means that if AB makes sense, then we can calculate justthe i-th column of AB without calculating all of AB. Namely, the i-th columnof AB is Acolumni(B).

Formally, we can define the product of a m by n matrix A with a n by kmatrix B to be the m by k matrix AB such that

(AB)i, j =n

∑l=1

(A)i,l(B)l, j

This formula looks difficult, but what it really tells us is is that the (i, j)entry of AB is really the dot product of the i-th row of A with the j-th column

Nicholas Long


of B. Remember the dot product of~v =

v1v2...

vn

∈Rn and ~w =

w1w2...

wn

∈Rn is

just the sum of the products of the components. Namely,

~v ·~w =n

∑i=1

viwi

This idea lets us calculate the matrix product AB one entry at a time. Con-tinuing this idea will lead us to see that the i-th row of the product AB canbe calculated as rowi(AB) = rowi(A)B.

Note that in general AB 6= BA, even when both products make sense.

Question 88. a) What sizes of matrices can you add to a m by n matrix?

b) What sizes of matrices can you multiply on the right of a m by n matrix?

c) What sizes of matrices can you multiply on the left of a m by n matrix?

Question 89. If A ∈Mm×n, when does it make sense to multiply by AT ?


[3 1−1 2

]and B =

[−1 2 31 0 −2

].

a) What is the size of AB?

b) Compute just the first column of AB.

c) Write the first column of AB as a linear combination of the columns ofA. Be sure to check your work.

d) Solve the matrix equation A~x =[−23

]e) Compute just the second row of AB


3 2 1 5 64 1 3 2 −10 2 5 6 78 2 3 1 4

and B =

5 −2 2 46 2 3 64 −1 7 142 0 −2 −41 1 2 4

a) (A)2,3 =

b) (B)1,4 =

Nicholas Long


c) (AB)2,3 =

d) row2(AB) =

e) column3(AB) =


[3 1−1 2

]and B =

[−1 2 31 0 −2

]. Compute AB and

BA.


3 1−1 2−2 0

and B =

[−1 2 31 0 −2

]. Compute AB and

BA.

Question 94. Prove that (A+ B)C = AC + BC for matrices A, B, and Csuch that the addition and multiplication of matrices makes sense. You canapproach this problem by showing matrix equality entry-wise or column-wise or row-wise.

Question 95. Give 2 different examples of 3 by 3 matrices A and B suchthat AB = BA.

Question 96. Give examples of 3 by 3 matrices A and B such that AB 6= BA.

Question 97. Prove (AB)T = BT AT .

Question 98. Matrices are good for efficiently storing information. Forinstance, the adjacency matrix of a directed graph works as follows:

• If A is n by n, the corresponding directed graph has n vertices.

• There are (Ai, j) edges from vertex i to vertex j.

For instance, if B =

[1 21 0

], then GB = • (( ��

99 •``

a) Draw the directed graph with adjacency matrix A =

1 2 32 1 00 2 4

b) Compute A2 = AA and describe what (A2)2,1 means in terms of GA, the

directed graph of A. What would (A2)1,2 or (A2)3,2 mean?

c) What does (An)i, j mean for n≥ 1?

Nicholas Long

Chapter 2

Vector Spaces

Vector spaces are the primary objects that we study in linear algebra. Gen-erally speaking, a vector space is a collection of objects (called vectors)that preserves linear relationships; that is the objects work well under vectoraddition and scalar multiplication. As you will see shortly, vectors are notalways going to be the column vectors of Chapter 1 or viewed geometricallyas arrows from one point to another.

Definition 99. A vector space, V , is a nonempty set of objects called vectorswith two operations called addition and scalar multiplication such that thefollowing hold for all~u,~v,~w ∈V and c,d ∈ R:

a) If~u,~v ∈V , then~u+~v ∈V .

b) ~u+~v =~v+~u

c) (~u+~v)+~w =~u+(~v+~w)

d) There exists a vector~0 such that~v+~0 =~v

e) For each~u ∈V , there is a vector −~u such that~u+(−~u) =~0.

f) If~u ∈V and c ∈ R, then c~u ∈V .

g) c(~u+~v) = c~u+ c~v

h) (c+d)~v = c~v+d~v

i) c(d~v) = (cd)~v

j) 1~v =~v

You can refer to these properties as

a) closure of vector addition

25

Vector Spaces 26

b) commutativity of vector addition

c) associativity of vector addition

d) existence of the zero vector

e) existence of the additive inverse

f) closure of scalar multiplication

g) distributive property of scalar multiplication across vector addition

h) distributive property of scalar addition across scalar multiplication (ofa vector)

i) associativity of scalar multiplication

j) existence of scalar multiplicative identity

This is the definition for a real vector space since the scalars come fromR, the real numbers. Sometimes it will be useful for us to consider complexvector spaces (scalars come from C), but unless otherwise stated, you canassume that you are working with a real vector space.

Question 100. In order to gain an appreciation of definitions, use just theabove definition to prove the following results:

a) The zero vector is unique. You can begin this by supposing that thereexists some ~w such that ~x+ ~w =~x for every ~x ∈ V , then you need toshow that ~w must equal~0V .

b) The additive inverse of a vector is unique.

Example 101. The real numbers, R, are a vector space since all of theabove properties hold.

Example 102. Real valued vectors, Rn, are a vector space since all of theabove properties hold when vector addition and scalar multiplication aredone componentwise. We can think of the vectors in Rn as a directed linesegment (an arrow) or as a point in n-dimensional space.

Question 103. Show why Zn, the set of n-vectors with integer componentsis not a vector space.

Question 104. Is Cn a real vector space? Why or why not?

Question 105. Is Rn a complex vector space? Why or why not?

Nicholas Long

Vector Spaces 27

Example 106. The set of m by n matrices over the real numbers, Mm×n(R)or simply Mm×n, is a vector space since all of the above properties holdwhen ”vector” addition and scalar multiplication are done entry wise. Thequotes are around vector in the previous sentence because you may not al-ways think of matrices as being vectors but using the properties from Section1.6, you can treat matrices as vectors in the general sense.

Question 107. Prove that Pn, the set of polynomials (in variable t) withdegree at most n, is a real vector space.

Example 108. The following sets are also vector spaces:

a) The set of all polynomials (in variable t) denoted P.

b) (R)Z, the set of bi-infinite sequences of real numbers. If x∈ (R)Z, thenx = ...x−3,x−2,x−1,x0,x1,x2,x3, ... where xi ∈R. Scalar multiplicationand vector addition are done coordinate-wise in this vector space, i.e.x+y = ...x−3+y−3,x−2+y−2,x−1+y−1,x0+y0,x1+y1,x2+x2,x3+y3, ....

c) F(S,R), the set of functions from a set S to the real numbers.

d) {~0}, the trivial vector space.

Question 109. We are going to take a look at the vector spaceV = F({a,b,c},R) to get used to our more general way to think about vec-tors and vector spaces.

• Let g1 be the function that takes a, b, and c to respectively.

• Let g2 be the function that takes a, b, and c to−2, 7, and 1 respectively.

• Let g3 be the function that takes a, b, and c to 1, 1, and 1 respectively.

• Let g4 be the function that takes a, b, and c to 0, 0, and 0 respectively.

a) Does it make sense to add the inputs of these functions? Explain.

b) Does it make sense to add the outputs of these functions? Explain.

c) Let g5 be the function that takes 5, 1, and 0 to a, b, and c respectively.Is g5 ∈V ?

d) Describe the function g1 +g2.

e) Describe the function 3g3.

f) What function when added to g2 will give g4?

Nicholas Long

Vector Spaces 28

g) Can you write g1 as a linear combination of the set {g2,g3,g4}? Ex-plain why or why not.

h) Can you write g4 as a linear combination of the set {g2,g3,g1}? Ex-plain why or why not.

Question 110. a) What is the zero vector for the vector space Mm×n?

b) What is the zero vector for the vector space P?

c) What is the zero vector for the vector space F(R,R)?

d) What is the zero vector for the vector space (R)Z?

2.1 Subspaces

As Question 107 shows, it can be very tedious to prove that a set is indeeda vector space. A subspace of a vector space is a subset that is itself avector space. Since most of the properties of the vector spaces we look atget inherited from some larger vector space, it is often easier to show thata set is a vector space by showing it is a subspace of the appropriate parentvector space.

Theorem 111. A subset H of a vector space V is a subspace if and only ifthe following are true:

a) The zero vector of V is in H;~0V ∈ H.

b) H is closed under vector addition; if~u,~v ∈ H, then~u+~v ∈ H.

c) H is closed under scalar multiplication; if~u ∈ H and c ∈ R, then c~u ∈H.

This theorem is so useful because we can prove a set is a vector spaceby checking only three properties instead of the ten that are involved in thedefinition. The reason that we do not need to check these other properties isthat by using this subspace, we already have the proper rules of arithmeticproven from the parent space. Additionally, since we are using the samerules for scalar multiplication and vector addition as the parent space, wemust also have the same scalars as the parent space.

Question 112. Use the preceding theorem to prove that Pn is a subspace ofP.

Question 113. Is R a subspace of C? Why or why not?

Nicholas Long

Vector Spaces 29

Question 114. Is R2 a subspace of R3? Why or why not?

Question 115. Is the set of points on the plane given by z = 0 a subspace ofR3? Why or why not?

Question 116. Is the set of points on the plane given by z = 1 a subspace ofR3? Why or why not?

Question 117. Draw a plot of the points in R2 given by {~x ∈R2|x1x2 ≥ 0}.

Is {~x =[

x1x2

]∈ R2|x1x2 ≥ 0} a subspace of R2? Why or why not?

Question 118. Is Symn×n, the set of symmetric n by n matrices a subspaceof Mn×n? Why or why not?

Question 119. Prove or disprove: The set of odd functions on R ( f (−t) =− f (t) for every t ∈ R) a subspace of F(R,R).Question 120. If A is a m by n matrix, prove that the solution set to thehomogeneous equation A~x =~0 is a subspace of Rn

Question 121. Prove that if H and K are subspaces of some vector spaceV , then the set H

⋂K is a subspace of V as well.

Question 122. Prove or Disprove: the set of 2 by 2 matrices with at leastone zero entry is a subspace of M2×2.

Question 123. Prove or Disprove: the set of matrices of the form[

a b0 −a

]is a subspace of M2×2.

Question 124. Prove or disprove: The set of quadratic polynomials of theform at2 +b is a subspace of the vector space of polynomials.

2.2 Span

Recall that a linear combination of the set {~v1, ..~vk} is a vector of the formn

∑i=1

ci~vi = c1~v1 + c2~v2 + ...+ ck~vk

Note that some of the ci may be zero. In other words, not every vector in aset needs to be part of a linear combination from that set.

Question 125. Can you write~b =

[24

]as a linear combination of ~v1 =

[11

]and ~v2 =

[−11


Nicholas Long

Vector Spaces 30

Question 126. Repeat the previous problem for ~b =

[00

], ~b =

[−12

], and

~b =

[22

].

Question 127. Can you write 2+ 4t as a linear combination of 1+ t and−1+ t? Justify your answer.

Question 128. Can you write[

2 31 1

]as a linear combination of

[1 10 1

]and

[1 2−1 1


The span of a set of vectors S, denoted span(S) is the set of all possiblelinear combinations of S. A set S is said to span or generate a vector spaceV if span(S) =V .

Question 129. If S =

1

11

, 1−11

, is~b =

313

∈ span(S)?


1

11

, 1−11

, is~b =

123

∈ span(S)?


1

11

, 1−11

, is~b =

000

∈ span(S)?


1

11

, 1−11

, 1−1−1

, is~b =

123

∈ span(S)?

Question 133. IS 1− t2 in the span of {3,4+ t + t2,5− t}? Be sure toproperly justify your answer.

Question 134. For what value(s) of α and β is ~p =

β

−2α

−4

a solution to

A~x =~b if A =

1 5 −2 0−3 1 9 −54 8 −1 7

and~b =

−790

?

Nicholas Long

Vector Spaces 31

Question 135. Is ~b =

−790

in the span of the set of columns of A = 1 5 −2 0−3 1 9 −54 8 −1 7

? If so, what are the coefficients?

Question 136. Prove that if S is a set of k vectors from a vector space V ,then span(S) is a subspace of V .

Question 137. Find a finite set of vectors that generates each of the follow-ing vector spaces (be sure to prove why your set works):

a) R3

b) P2

c) Symn×n

Question 138. Show that the set {1+ t, t + t2,1+ t3, t + t2 + t3} spans allof P3. Hint: Come up with a system of equations that you will need to solveand use your theorems from Chapter 1.

Question 139. Geometrically describe the span of

2

14

.

Question 140. Geometrically describe the span of

2

14

, 3−11

.

Question 141. Does the span of

2

14

, 3−11

have to go through the

origin? Justify your answer.

Question 142. Does the span of S = {~v1, ...,~vk} where ~vi ∈ Rn have to gothrough the origin? Justify your answer.

2.3 Linear Independence

Definition 143. A set of vectors S is linearly independent if the only linearcombination of the zero vector is the trivial linear combination. In otherwords, S being a linear independent set implies that if c1~v1 + c2~v2 + ...+ck~vk =~0 where ~vi ∈ S, then all ci = 0.

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Vector Spaces 32

A set of vectors S is linearly dependent if the set is not linearly depen-dent. More specifically, there exists a solution to c1~v1+c2~v2+ ...+ck~vk =~0where ~vi ∈ S and at least one of the c j 6= 0.

Question 144. Is the set

1−32

linearly independent? Justify your an-

swer.

Question 145. Is the set

2

30

,−1−12

linearly independent? Justify

your answer.

Question 146. a) Choose a vector~v so that the set

2

30

,−1−12

,~v is

linearly independent, where~v ∈ R3.

b) Is your choice of~v in Span

2

30

,−1−12

,~v?

Question 147. Is {2+ t2,1+ t2} a linearly dependent set in P2?

Question 148. Is{[

1 10 0

],

[0 01 1

],

[1 00 1

],

[0 11 0

]}a linearly indepen-

dent set in M2×2?

Question 149. Prove that {1+ t, t + t2,1+ t2} is linearly independent.

Question 150. Prove that if a set S of a vector space V contains~0V , then Sis linearly dependent.

Question 151. If A is a m by n matrix, then the columns of A form a linearlyindependent set if and only if A has pivot columns. Completelyjustify your response.

Question 152. Prove the following statement: If M = {~v1,~v2, ...,~vn} is lin-early independent, then any subset of M is linearly independent.

Question 153. Prove or disprove: If M = {~v1,~v2, ...,~vn} is linearly depen-dent, then any subset of M is linearly dependent.

Question 154. Prove that if ~u is in the span of S, then S⋃{~u} is linearly

dependent.

Nicholas Long

Vector Spaces 33

The following two questions are a wonderful summary of the differencebetween and the importance of linear dependence and linear independence.

Question 155. Prove that if S is a linearly dependent set, then any ~w ∈span(S) can be written as a linear combination from S in more than oneway.

Question 156. Prove that if S is a linearly independent set, then any ~w ∈span(S) can be written as a linear combination from S in only one way.

2.4 Linear Transformations

Linear transformations are the “nice”functions from vector spaces to vectorspaces. In particular, linear transformations preserve the operations of scalarmultiplication and vector addition.

Definition 157. A function T from a vector space V to a vector space W isa linear transformation if for every ~v1,~v2 ∈V and c ∈ R

• T (~v1 +~v2) = T (~v1)+T (~v2)

• T (c~v1) = cT (~v1)

Question 158. Prove that the map T : Rn → Rm given by T (~x) = A~x islinear, where A is an m by n real values matrix.

Eventually we will be able to state a lot of linear transformations as amatrix transformation like in the problem above, but we will not be ableto do this in general.

Question 159. Prove that the map T : P→ P given by T ( f ) =d fdt

is linear.

Question 160. For each of the following functions, determine if the functionis a linear transformation:

a) f1 : P→ R where f (~p) = the degree of the polynomial ~p

b) f2 : P→ R where f (~p) = ~p(t = 1)

c) f3 : R2→ R3 where f ([

ab

]) =

a+ba−bb+1

.

d) f4 : R3→ R2 where f (

abc

) = [a+ba− c

].

Nicholas Long

Vector Spaces 34

e) f5 : R3→ R2 where f (

abc

) = [a+bc2

].

Question 161. Prove that if T is a linear transformation and a set of vec-tors {v1,v2,v3} is linearly dependent, then the set {T (v1),T (v2),T (v3)} islinearly dependent.

Question 162. Give a counterexample to the following statement: If T isa linear transformation and a set of vectors {v1,v2,v3} is linearly indepen-dent, then the set {T (v1),T (v2),T (v3)} is linearly independent.

Question 163. Prove that if T is a linear transformation from V to W, thenT (~0V ) =~0W .

Question 164. If a linear transformation, T , sends the vector ~e1 =

[10

]to 3

−11

and sends the vector ~e2 =

[01

]to

102

, find the following:

• T([

30

])• T

([05

])• T

([ab

])Question 165. Find a matrix A such that for the transformation in the pre-vious problem T (~x) = A~x.

If T is a linear transformation from Rn to Rm, then the standard matrixpresentation of T is a m by n matrix

A = [T (~e1) T (~e2) ... T (~en)]

where ~ei is the i-th elementary vector of Rn. Note that (~ei) j = δi, j, whereδ is the Dirac delta function. Recall that

δi, j =

{0 if i 6= j1 if i = j

The vector ~ei can also be thought of as the i-th column of Idn, the n by nidentity matrix. Because of how we defined the standard matrix presenta-tion, only transformations from Rm to Rn will have standard matrix presen-tations. In particular, the standard matrix presention keeps track of wherethe standard basis vectors (~ei) go under the transformation T .

Nicholas Long

Vector Spaces 35

Question 166. Write out ~e1, ~e2, and ~e3 from R3. What is the result of mul-

tiplying

a b cd e fg h i

by ~e1? What about ~e1? ~e3?

What would this mean for the following matrix product:a b cd e fg h i

[~e1 ~e2 ~e3]

Question 167. Determine the standard matrix presentation A for the fol-lowing T :

• T : R2→ R2 reflects points over the vertical axis.

• T : R2→ R2 rotates points clockwise by π/2.

• T : R2→ R2 rotates points by π and then flips points over the verticalaxis.

Question 168. Draw what the image of the picture below will look like afterapplying given the linear transformations

a) T : R2→ R2 reflects points over the vertical axis.

b) T : R2→ R2 rotates points clockwise by π/2.

c) T : R2→ R2 where T (~x) = A~x and A =

[1 20 1

]d) T : R2→ R2 where T (~x) = A~x and A =

[3 00 2

]

Nicholas Long

Vector Spaces 36

e) T : R2→ R2 where T (~x) = A~x and A =

[0 −11 0

]f) T : R2→ R2 where T (~x) = A~x and A =

[0 00 1

]Question 169. If a linear transformation, T , sends the vector

[11

]to[−22

]and sends the vector

[−11

]to[

02

], find the following:

• T([

10

])Hint: How can you write

[10

]as a linear combination of

[11

]and

[−11

].

• T([

01

])• T

([ab

])• Find the standard matrix presentation for T

Question 170. Let T~0 be the function from V to W such that T (~x) =~0W forevery~x ∈ V . Let IdV be the identity map on V , IdV (~x) =~x for every~x ∈ V .Prove that T~0 and IdV are linear.

The range of a linear transformation T : V →W is the set of things inthe codomain W that are the output of T for some input. That is range(T ) ={~y ∈W |~y = T (~x) for some~x ∈ V}. The null space, or kernel, of a lineartransformation T : V →W is the set of inputs that get mapped to the zerovector of W . That is Null(T ) = {~x ∈V |T (~x) = ~0W}.

Question 171. Is~b =

021

in the range of the linear transformation T (~x) =

A~x where A =

1 23 40 0

? Justify your answer without doing any matrix oper-

ations. Hint: write the corresponding matrix equation as a vector equation.


[1 2 34 5 6

]. Find the range and null space of T

where T (~x) = A~x.

Nicholas Long

Vector Spaces 37

Question 173. Let T from R2 to P2 be given by T([

ab

])= a+(a+b)t +

(a−b)t2.

a) Prove T is linear.

b) Compute the range of T .

c) Compute the null space of T .

Question 174. Let V be the vector space of polynomials in x and y.

a) Show the transformation T that maps f to∂ f∂x

is a linear transforma-tion.

b) Compute the null space of T .

c) Compute the range of T .

Question 175. Let T be a linear transformation from V to W. Prove thatnull(T ) is a subspace of V .

Question 176. Let T be a linear transformation from V to W. Prove thatrange(T ) is a subspace of W.

A function f : C→D is one to one if whenever f (x) = f (y), then x = y.This means that each input gets sent to a different output by the function f .Alternately, you can say a one to one function does not map two differentinputs to the same output.

A function f : C→ D is onto if every element of D has some input thatis mapped to it. In other words, the range of f is all of D.

Question 177. For each of the functions from R to R below state whetherthe function is either 1-1 but not onto, onto but not 1-1, 1-1 and onto, or not1-1 and not onto.

a) f (x) = ex

b) f (x) = x

c) f (x) = x2

d) f (x) = 1− x

e) f (x) = x2(1− x)

f) f (x) = sin(x)

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Vector Spaces 38

g) f (x) = x3


ab

])= a+(a+b)t +

(a−b)t2.

a) Is T one-to-one?

b) Is T onto?

Question 179. Give an example of a linear transformation from R2 to R3

that is one to one.


that is onto.


that is onto.

Question 182. If the set of columns of a m by n matrix A are linearly inde-pendent, does the set of columns of A span all of Rm?

Question 183. If the set of columns of a m by n matrix A are linearly inde-pendent, is the range of T (~x) = A~x all of Rm?

Theorem 184. A linear transformation T :V →W is onto iff range(T )=W.

Question 185. Prove that for T a linear transformation from V to W,null(T ) = {~0} iff T is 1-1.

2.5 Applications

Definition 186. Let C n(R,R), or simply C n be the set of functions from Rto R that are n times continuously differentiable.

Question 187. A solution to the system of differential equations:

dxdt

= a1 x(t)+a2 y(t)

dydt

= a3 x(t)+a4 y(t)

is a choice of x and y as functions of t such that both differential equationsare satisfied. Prove that the set of solutions to the system above is a subspaceof the vector space (C 2)2 = {(x(t),y(t))|x,y ∈ C 2}.

Nicholas Long

Vector Spaces 39

The previous result is especially important in a differential equationsclass because finding the solution set of the system of differential equationsreduces to finding a few solutions that spans a large enough space.

Question 188. a) Let S be the set of solutions to the differential equation

ad2xdt2 +b

dxdt

+cx(t) = 0. Prove that S = { f ∈ F(R,R)|ad2 fdt2 +b

d fdt

+

c f (t) = 0} is a subspace of F(R,R).

b) If f1(t) and f2(t) are solutions to the differential equation ad2xdt2 +

bdxdt

+c(x(t)) = g(t), then prove that f1− f2 is a solution to the homo-

geneous differential equation ad2xdt2 +b

dxdt

+ c(x(t)) = 0.

c) Conclude that the solution set of the non-homogeneous differentialequation is of the form y(t)+ S, where y is a solution to the nonho-mogeneous differential equation and S is the solution set to the homo-geneous differential equation.

The previous problem is analogous to your work on Question 50.

Question 189. a) Show that the transformation T from C 2 to C 2 given

by T ( f ) = ad2 fdt2 +b

d fdt

+ c f (t) is linear.

b) What is Null(T )?

Nicholas Long

Chapter 3

Connecting Ideas

3.1 Basis and Dimension

In the previous chapter, we saw that if a set S spans a vector space V , thenS is big enough to write everything in V (as a linear combination of S). Wealso saw that a linearly independent set S had a unique way to representelements in span(S) (Question 156). A convenient and minimal way todescribe a vector space is to give a set of vectors that spans all of V but doesnot include anything extra.

Definition 190. A basis for a vector space V is a set of vectors that is lin-early independent and spans V .

In this way, a basis is big enough (spans V ) and nothing extra (linearlyindependent).

Question 191. Can a set of 4 vectors be a basis for R3? Why or why not?Be sure to justify using ideas from Chapter 1 and not any theorems past thispoint.

Question 192. Can a set of 2 vectors be a basis for R3? Why or why not?

Theorem 193. If there exists a basis for a vector space V with n vectors,then every basis of V must have exactly n vectors.

The previous theorem does not imply that there is only one basis for avector space, just that any two bases for the same vector space will havethe exact same number of vectors. The idea that every basis for a vectorspace V has the same number of vectors gives rise to the idea of dimension.Namely, if a vector space V has a basis with a finite number of elements, n,then we say that V has dimension n or that V is n-dimensional, also writtenas dim(V ) = n.

40

Connecting Ideas 41

Question 194. Show that {~e1, ...,~en} is a basis for Rn and thus that Rn isan n-dimensional vector space.

Question 195. Give a set of 3 different vectors in R3 that are not a basis forR3. Be sure to show why the set does not satisfy the definition of a basis.

Question 196. Give a basis for P3 and compute the dimension of P3.

Question 197. What is dim(Pn)?

Question 198. Recall that the set {~0} is the trivial vector space. What is abasis for {~0}? What is dim({~0})?

Theorem 199. If V is an n-dimensional vector space and S is a set withexactly n vectors, then S is linearly independent if and only if S spans V .

This is an enormously helpful theorem since we know that a linearlyindependent set of n vectors from a n-dimensional vector space is a basis(no need to show spanning). This goes the other way as well, namely if aset of n vectors, S, spans a n-dimensional vector space, V , then S is a basisfor V (no need to show linear independence).

Question 200. Prove that {1+ t, t + t2,1+ t2} is a basis for P2.

Question 201. Give two different bases for M2×2.

Question 202. What is the dimension of Sym3×3, the vector space of sym-metric 3 by 3 matrices?

Question 203. What is the dimension of P? Justify your answer.

Question 204. a) Prove that H = {

tt0

|t ∈ R} is a subspace of R3.

b) Is Span({

100

,0

10

}) = H?

c) What dimension is H?

3.1.1 rank and nullity

Recall from Question 175, if T : V → W is linear, then Null(T ) andrange(T ) are subspaces of V and W respectively. The rank of a trans-formation T is dim(range(T )) and the nullity of T is dim(Null(T )).

Nicholas Long

Connecting Ideas 42


[1 2 34 5 6

].

a) Find rank(T ) and nullity(T ) where T (~x) = A~x.

b) Find a basis for Null(T ).

c) Find a basis for range(T ).


ab

])= a+(a+b)t +

(a−b)t2.

a) rank(T ) =

b) nullity(T ) =

c) Find a basis for Null(T ).

d) Find a basis for range(T ).

Question 207. Let T : P3 → R2 be given by T ( f ) =[

f (0)f (1)

]. Compute

rank(T ) and nullity(T ).

Theorem 208 (Dimension Theorem). Let T be a linear transformation fromV to W with V a n-dimensional vector space. rank(T )+nullity(T ) = n.

If T is a matrix transformation (T (~x) = A~x), then the rank(T ) =rank(A) = dim(Col(A)) and nullity(T ) = nullity(A) = dim(Null(A)).

Question 209. Using Question 63 and other previous work, prove the Di-mension Theorem for T : Rn→ Rm given by T (~x) = A~x, where A is a m byn matrix.

Question 210. List out all possible echelon forms of 3 by 3 matrices usingthe symbols� for pivot, ∗ for non-pivot entry (possibly 0) , and 0 if an entrymust be 0. For each form, give the rank of the matrix and the dimension ofthe null space.

3.1.2 Coordinate Vectors relative to a basis

Given an ordered basis β = {~v1, ...,~vk} of a vector space V , the coordi-nate vector of ~x relative to β , denoted [~x]β , is a vector of the coefficients

Nicholas Long

Connecting Ideas 43

of the unique way to write ~x as a linear combination of β . Namely, if

~x = c1~v1 + c2~v2 + ...+ ck~vk, then [~x]β =

c1c2...

ck

.

Question 211. For each of the following vectors, compute [v]β where β =

{

001

,0

11

,1

11

}a) v =

222

b) v =

300

c) v =

−1−10

d) v =

−203

e) v =

abc

Question 212. For each of the following vectors, compute [v]β where β =

{1+ t, t + t2,1+ t2}

a) v = 2+2t

b) v = 4− t2 + t

c) v = 3

d) v = t

e) v = 6t2

The coordinate vector allows us to state problems in a vector space likePn in the same way we state problems in Rn.

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Connecting Ideas 44

3.2 Invertible Matrices

In this section, we will only consider square matrices. A matrix A ∈Mn×nis invertible if there exists a matrix B such that AB = Idn and BA = Idn.The inverse matrix of A is denoted A−1. Be careful that you do not use thenotation A−1 until you have shown that A is invertible.

3.2.1 Elementary Matrices

Recall that an elementary row operation on a matrix is an operation of theform:

• multiplying a row by a non-zero scalar

• switching two rows

• adding a multiple of one row to another row

Elementary matrices are obtained by performing an elementary operationon the identity matrix.

Question 213. Give the elementary matrix obtained by performing the givenoperation on Id3:

• Scaling the first row by α

• Switching the second and third rows

• Adding 3 times the 2nd row to the 1st row.

• Adding 3 times the 1st row to the 2nd row.

Question 214. Check that your answer to the previous question does the de-

sired operation by multiplying each of the elementary matrices by

a b cd e fg h i

.

Which side do you multiply the elementary matrix on to correspond to rowoperations?

Question 215. Compute (and verify) the inverse of each of the elementarymatrices from the previous problems. (Hint: Think about how you would gobackwards for each of the elementary operations.

Your work on the previous questions should convince you that elemen-tary matrices are invertible and that multiplying by an elementary matrix

Nicholas Long

Connecting Ideas 45

produces the same result as having performed the corresponding elemen-tary row operation. Elementary matrices offer a way of keeping track ofelementary operations.

Theorem 216. Elementary matrices are invertible and the inverse matrix isan elementary matrix corresponding to the inverse elementary operation.

Theorem 217. If A and B are invertible n by n matrices, then AB is aninvertible n by n matrix. Further, (AB)−1 = B−1A−1.

Question 218. Prove that if A can be reduced to Idn by elementary rowoperations, then A is invertible.

Question 219. Give all values of k where A =

1 0 2−1 k 43 5 1

will be invert-

ible.

Question 220. Give all values of k where A =

1 0 2−1 k 43 −1 1

will be in-

vertible.

Question 221. How many pivots must a matrix A have in order to be rowreducible to Idn? Justify using previous results.

Question 222. Prove that if A is invertible, then A~x =~b has a unique solu-tion for every~b ∈ Rm.

Question 223. Prove or disprove: If A and B are invertible n by n matrices,then A+B is invertible.

Question 224. Prove that if A is invertible, then AT is invertible.

3.2.2 Computing Inverses

In general computing the inverse of a matrix takes more time and operationsthan solving a system of equations. For this reason, it is generally easier tofind and solve a related system of equations problem than to compute theinverse matrix. We will outline a few ways to find inverse matrices andcompute a few small examples.

Question 225. If a matrix A is row reduced to Idn by elementary row oper-ations corresponding (in order of use) to elementary matrices E1, E2, ... ,Ek, give an expression for A−1.

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Connecting Ideas 46

Question 226. Use your answer to the previous question to prove the fol-lowing:

Any sequence of elementary row operations that reduces A to Idn alsotransforms Idn into A−1.

The previous result shows that computing inverses is equivalent to a rowreduction problem. In particular, if A is invertible, then reducing [A | Idn]to reduced row echelon form will produce the matrix [Idn | A−1].

Question 227. Use the idea above to compute the inverse of[

a bc d

]. Be sure

to note any assumptions you will need to make in order to reduce [A | Idn]to [Idn | A−1].

Question 228. If A =

1 0 10 2 −10 6 −1

, find A−1 and check that AA−1 = Id3.

Question 229. If A =

[0 −13 4

], find A−1 and use your answer to solve

A~x =~b if:

a) ~b =

[31

]b) ~b =

[−1−2

]c) ~b =

[05

]d) ~b =

[α

β

]

3.2.3 Invertible Matrix Theorem

Question 230. In many texts there is a long list of equivalent conditionsfor when a square matrix is invertible. Below is a list of some of theseconditions that we have talked about or proven. Go back through your notesand questions and cite when we connected two of the ideas in the list. Forinstance, parts a) and b) are linked by Question 218

Theorem 231 (The Invertible Matrix Theorem). Let A be a n by n matrix.The following are equivalent statements (either all True or all False):

Nicholas Long

Connecting Ideas 47

a) A is an invertible matrix.

b) A is row equivalent to Idn.

c) A has n pivots.

d) rank(A) = n

e) nullity(A) = 0

f) A~x =~0 has only the trivial solution.

g) The linear transformation~x→ A~x is one-to-one.

h) The linear transformation~x→ A~x is onto.

i) A~x =~b has a solution for every~b ∈ Rn.

j) The columns of A form a linearly independent set.

k) The columns of A span Rn.

l) The columns of A are a basis for Rn.

m) AT is invertible.

3.3 Invertible Linear Transformations

Definition 232. A linear transformation T from V to W is invertible if thereexists a linear transformation U from W to V such that T ◦U = IdW andU ◦T = IdV .

Alternative definition: A linear transformation T from V to W is invert-ible if T is one-to-one and onto.


[1 2 34 5 6

]. Is T : R3→ R2 given by T (~x) = A~x an

invertible linear transformation?


ab

])= a+(a+b)t +

(a−b)t2. Is T an invertible linear transformation?

Question 235. Let T from R3 to P2 be given by T

abc

= (a+c)+(a+

b)t +(a−b)t2. Is T an invertible linear transformation?

Nicholas Long

Connecting Ideas 48

3.4 LU factorization of matrices

Question 236. a) Reduce A =

[1 23 4

]to echelon form. How many row

operations did you use?

b) Compute the elementary matrix (or matrices) corresponding to the rowoperation(s) from the previous problem.

c) Compute the inverse of the elementary matrix from the previous prob-lem and multiply the result by the echelon form you found in part b).What is your result and why does this make sense?

d) Let L be the inverse elementary matrix from part c) and let U be the

echelon form from part a). Solve L~y =~b for~b =

[1−1

].

e) Now solve U~x =~y.

f) Now solve A~x =~b.

Question 237. a) Reduce A =

1 2 34 5 67 8 9

to echelon form. How many

row operations did you use?

b) Compute the elementary matrix (or matrices) corresponding to the rowoperation(s) from the previous problem.

c) Compute the inverse of the elementary matrix (or product of matrices)from part b) and multiply your answer by the echelon form you foundin part a). What is your result and why does this make sense?

d) Let L be the inverse elementary matrix product from part c) and let U

be the echelon form from part a). Solve L~y =~b for~b =

1−10

.

e) Now solve U~x =~y.

f) Now solve A~x =~b.

The preceding two problems can be generalized to show how row oper-ations will conveniently reduce any matrix into a product of an upper- anda lower-diagonal matrix. This LU decomposition has certain advantageswhen solving linear systems using a computer, especially for large systems.

Nicholas Long

Connecting Ideas 49

3.5 Determinants

Determinants will be an incredibly useful tool in quickly determining sev-eral important properties of square matrices. We will first look at how tocompute determinants and later outline the important properties that deter-minants have. While some of you may have been taught some rules for howto compute determinants of 2 by 2 and 3 by 3 matrices, I encourage you tounderstand how to compute determinants in general.

3.5.1 Computing Determinants

Definition 238. The determinant is a function from n by n matrices to thereal numbers (det : Mn×n → R). If A is a 1 by 1 matrix A = [A1,1], thendet(A) = A1,1. For n≥ 2, the determinant of a n by n matrix is given by thefollowing formula in terms of determinants of (n−1) by (n−1) matrices:

det(A) =n

∑j=1

(−1)1+ j(A)1, j det(A∗1, j)

where A∗i, j is the (n−1) by (n−1) matrix obtained by deleting the i-th rowand j-th column of A.

The term (A)i, j det(A∗i, j) is called the (i, j) cofactor of A.The range of the determinant is C if the entries of the matrix are complex.

The above definition uses cofactor expansion along the first row.

Question 239. In this question, we will unpack the determinant formula

above for a 2 by 2 matrix A =

[a bc d

].

a) Rather than using the summation notation of the formula above, writeout the two terms in det(A).

b) A∗1,1 =

c) A∗1,2 =

d) (A)1,1 =

e) (A)1,2 =

f) (−1)1+1 =

g) (−1)1+2 =

Nicholas Long

Connecting Ideas 50

h) det(A) =

Your answer to the previous problem will be useful in calculating deter-minants of 3 by 3 matrices.

Theorem 240. The determinant can be computed by cofactor expansionalong any row or column. Specifically the cofactor expansion along the k-throw is given by

det(A) =n

∑j=1

(−1)k+ j(A)k, j det(A∗k, j)

and the cofactor expansion along the k-th column is given by

det(A) =n

∑i=1

(−1)i+k(A)i,k det(A∗i,k)

Question 241. Use cofactor expansion along the first column of

A =

a b cd e fg h i

to compute det(A).

Question 242. Use cofactor expansion along the second row of

A =

a b cd e fg h i

to compute det(A). Did you get the same answer as the

previous question?

Question 243. Compute the determinant of B =

g h id e fa b c

. How does

your answer compare with the previous problem.

Question 244. Compute the determinant of C =

a b c3d 3e 3 fg h i

.

Question 245. Compute the determinant of D=

a+ kd b+ ke c+ k fd e fg h i

.

Question 246. Compute the determinant of the following matrices:

a)

3 0 1 00 2 −1 4−3 5 0 22 2 2 −1

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Connecting Ideas 51

b) 2

3 0 1 00 2 −1 4−3 5 0 22 2 2 −1

3.5.2 Properties of Determinants

Question 247. Prove that det(Idn) = 1.

Question 248. Use your results from the previous subsection to state whatthe determinants of the different kinds of elementary matrices are. (You donot need to prove the general case, just give a clear, correct statement.)

Question 249. Prove that if A has a row of zeros, then det(A) = 0.

Theorem 250. a) If A and B are n by n, then det(AB) = det(A)det(B).

b) The determinant of an upper or lower triangular matrix is the productof its diagonal entries.

det(L) =n

∏i=1

Li,i

det(U) =n

∏i=1

Ui,i

c) The determinant of a diagonal matrix is the product of its diagonalentries. If D is diagonal, then

det(D) =n

∏i=1

Di,i

d) det(A) = det(AT )

e) A matrix A is invertible iff det(A) 6= 0.

Question 251. Let A = E1E2E3 be a four by four matrix, where

• E1 is the elementary matrix that adds two times the 2nd column to the3rd column

• E2 is the elementary matrix that switches the second and fourth rows

• E3 is the elementary matrix that scales the first row by 12

Compute det(A).

Nicholas Long

Connecting Ideas 52

Question 252. Let A~x = LU~x = ~b, where L =

1 0 0−2 1 02 −3 1

,

U =

3 −1 20 1 −20 0 5

, and ~b =

111

. Compute det(L), det(U),

and det(A). What relationship should these determinants have?


1 2 34 5 60 1 −1

and~b =

111

.

a) Compute det(A).

b) Let A1 be the matrix obtained by replacing the first column of A with~b. Compute det(A1).

c) By similar replacement of the second and third columns, find A2 andA3. Compute det(A2) and det(A3).

d) Solve A~x =~b.

e) Write

x1x2x3

in terms of det(A1), det(A2), det(A3), and det(A).

The previous problem is an example of Cramer’s Rule, which allows youto write the unique solution of A~x =~b (for a square invertible matrix A) interms of determinants.

Question 254. Prove: det(A) = 0 iff A~x =~0 has solutions such that~x 6=~0.

3.6 Eigenvalues and Eigenvectors

Definition 255. An eigenvector of a matrix A is a nonzero vector ~x suchthat A~x = λ~x for some scalar λ . The scalar λ is called an eigenvalue of Aif there exists a nonzero solution to A~x = λ~x.

Question 256. Which of the following vectors are an eigenvector of A =[2 33 2

]? For any vectors that are eigenvectors of A, give the eigenvalue.

a) ~v1 =

[12

]Nicholas Long

Connecting Ideas 53

b) ~v2 =

[−11

]c) ~v3 =

[3−1

]d) ~v4 =

[11

]e) ~v5 =

[00

]Question 257. Let A=

[2 1−1 3

]. Try to find an eigenvector with eigenvalue

3. In other words, find a vector~v such that A~v = 3~v.

Question 258. Let A=

[3 43 −1

]. Try to find an eigenvector with eigenvalue

−3. In other words, find a vector~v such that A~v =−3~v.

Question 259. Prove: det(A−αId) = 0 iff α is an eigenvalue. Hint: Lookat Question 254.

If A is a n by n matrix, then det(A− tId) will be a n-th degree polynomialin t, which we call the characteristic polynomial of A. The previous ques-tion shows that finding roots of the characteristic polynomial is the same asfinding eigenvalues.

Question 260. Find each of the following matrices: write out the charac-teristic polynomial, give all eigenvalues, and for each eigenvalue, find aneigenvector.

a)[

1 11 1

]b)[

1 −3−3 1

]c)[

1 23 4

]

d)

1 2 34 5 67 8 9

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Connecting Ideas 54

e)

4 −1 62 1 62 −1 8

f)

1 1 0 01 1 0 00 0 1 −30 0 −3 1

A root α of a polynomial (in t) has (algebraic) multiplicity k if k is the

largest integer such that (t−α)k is a factor.

Question 261. Prove that a nonzero vector, ~v, is an eigenvector of A witheigenvalue λ if and only if~v is in the null space of A−λ Id.

Question 262. Prove that if ~v is an eigenvector of A, then α~v is also aneigenvector of A.

Question 263. Prove that if ~v1 and ~v2 are eigenvectors of A with the sameeigenvalue, then ~v1 +~v2 is also an eigenvector of A. What is the eigenvalueof ~v1 +~v2?

If λ is an eigenvalue of A, then the eigenspace of λ , Eλ , is the set ofvectors~x such that (A−λ Idn)~x =~0. The previous two questions along withthe inclusion of~0 give the following theorem.

Theorem 264. If λ is an eigenvalue of A ∈Mn×n, then Eλ is a subspace ofRn.

Question 265. Prove that dim(Eλ )≥ 1 for every eigenvalue λ .

Question 266. a) Let A =

2 a b0 2 c0 0 2

. Show that A only has an eigen-

value of 2. What is the algebraic multiplicity of the eigenvalue 2?

b) Can you pick a, b, and c, so that the eigenspace of 2 has dimension 3?If so, give a choice of a, b, and c that does so.

c) Can you pick a, b, and c, so that the eigenspace of 2 has dimension 2?If so, give a choice of a, b, and c that does so.

d) Can you pick a, b, and c, so that the eigenspace of 2 has dimension 1?If so, give a choice of a, b, and c that does so.

Nicholas Long

Connecting Ideas 55

3.6.1 Diagonalizability

Definition 267. A matrix A is diagonalizable if there exists an invertiblematrix Q such that A = Q−1DQ where D is a diagonal matrix.

Theorem 268. A matrix A ∈ Mn×n is diagonalizable iff A has n linearlyindependent eigenvectors. In fact, the matrix Q that will diagonalize A willhave the n linearly independent eigenvectors as its columns.

The question becomes when can we find n linearly independent eigen-vectors for a matrix A.

Question 269. Can you diagonalize A =

[−1 2−2 4

]? If so, give a basis of

eigenvectors.

Question 270. Can you diagonalize A =

[1 −11 1

]? If so, give a basis of

eigenvectors.

Question 271. Prove that if ~v1 is an eigenvector with eigenvalue λ1 and ~v2is an eigenvector with eigenvalue λ2 6= λ1, then {~v1,~v2} is linearly indepen-dent.

The following theorem relies on the preceding question and the fact thatthe dimension of every eigenspace is at least 1.

Theorem 272. If a n by n matrix A has n distinct eigenvalues, then A isdiagonalizable.

The converse of this theorem is not true in that there diagonalizable ma-trices that do not have distinct eigenvalues.

Theorem 273. A n by n matrix A is diagonalizable iff the sums of the di-mensions of its eigenspaces is n.

Question 274. Give an example of a matrix that is not diagonalizable. Jus-tify your claim.

Question 275. Let A be a 4 by 4 matrix.

a) How many eigenvalues can A have?

b) For each of the possible number of eigenvalues in the previous part,write out all of the possible dimensions of each of the eigenspaces. Forinstance: if A has 4 distinct eigenvalues, then the only possibility isthat each eigenspace has dimension 1 (why is that?).

c) Which of the cases from the previous problem correspond to A beingdiagonalizable?

Nicholas Long

Connecting Ideas 56

3.6.2 Eigenvalues and Eigenvectors of Linear Transformations

The structure of eigenvalues, eigenvectors, and even diagonalizability canbe generalized to linear transformations if we consider a square matrix Aas a transformation~x→ A~x.

Definition 276. An eigenvector of a linear transformation T : V → V is anonzero vector~x ∈V such that T (~x) = λ~x for some scalar λ . The scalar λ

is called an eigenvalue of T if there exists a nonzero solution to T (~x) = λ~x.

Question 277. Examine each of the following transformations of R2 geo-metrically and find all eigenvalues and eigenvectors of the transformation.Hint: think about what kinds of vectors will be mapped to a scalar multipleof themselves.

a) T1 flips points over the horizontal axis.

b) T2 flips points over the line y = mx.

c) T3 rotates points by π counterclockwise.

d) T4 rotates points by π

3 counterclockwise.

e) T5 shears points horizontally by 2. In other words, T5(~e1) =

[10

]and

T5(~e2) =

[21

].

f) T6 projects points onto the vertical axis.

Question 278. What are the eigenvalues and eigenvectors of the transfor-

mation T : P→ P given by T ( f ) =d fdt

?

Question 279. Let T be the transformation of R2 given by T (~x) = A~x with

A =

[0 2−2 0

]. Describe geometrically what the linear transformation T

does.

Question 280. What are the eigenvalues and eigenvectors of A=

[0 2−2 0

]?

You need to consider complex numbers for both the eigenvalues and eigen-vectors. Be sure to check your eigenvalues and eigenvectors.

Question 281. What do you think the scalar multiplication by 2i is doing inthe previous problem?

Nicholas Long

Connecting Ideas 57

Question 282. Let T be the linear transformation from R2 to R2 that rotatesaround the origin by π

2 clockwise and then scales vectors by a factor of 2.Find A, the standard matrix for T and determine if A is diagonalizable.

Question 283. Let T be the linear transformation from R2 to R2 that rotatesaround the origin by θ counterclockwise. Find A, the standard matrix for T(in terms of θ ).

Determine for which values of θ A is diagonalizable.

Nicholas Long

Chapter 4

Inner Product Spaces

4.1 Inner Products

Recall the dot product of ~v =

v1v2...

vn

∈ Rn and ~w =

w1w2...

wn

∈ Rn is just the

sum of the products of the components. Namely,

~v ·~w =n

∑i=1

viwi =~vT~w

The dot product of a vector~x ∈ Rn with itself gives the length of the vectorsquared, ~x ·~x = ||~x||2. The dot product is the familiar example of an innerproduct on a real vector space.

In general, an inner product on a vector space V is a function from V×Vto R for real vector spaces (C for complex vector spaces), denoted by 〈∗,∗〉,such that for all~x,~y,~z ∈V and c ∈ R (or C):

a) 〈~x,~y〉= 〈~y,~x〉 (or 〈~x,~y〉= 〈~y,~x〉 for C).

b) c〈~x,~y〉= 〈c~x,~y〉 and 〈~x+~z,~y〉= 〈~x,~y〉+ 〈~z,~y〉

c) 〈~x,~x〉 ≥ 0.

Example 284. a) Rn with the dot product defined above is an inner prod-uct space.

b) C([0,1]), the set of continuous functions on the interval [0,1] is aninner product space when

〈 f ,g〉=∫ 1

0f (t)g(t) dt

58

Inner Product Spaces 59

.

c) Frobeinus Inner Product on Matrices

Two vectors~x and~y in an inner product space are orthogonal if 〈~x,~y〉=0.

Question 285. Find 3 different vectors that are orthogonal to[

12

].

Question 286. Find 3 different vectors that are orthogonal to

12−1

.

Question 287. Find a vector in C([0,1]) that is orthogonal to f (t) = t.

For vectors in Rn, the projection of ~x onto ~y computed as pro j~y~x =(~x·~y~y·~y

)~y

integral as inner product Frobenius inner product

4.2 Orthogonal Complements

A set of vectors is orthogonal if every pair of distinct vectors in the set isorthogonal.

Let W be a subspace of an inner product space V . The orthogonal com-plement of W, denoted W⊥, is the set of vectors in V that are orthogonal toevery vector in W.

Question 288. Prove that W⊥ is a subspace of V

Question 289. Let H = span({

100

,0

10

}. What is H⊥?

Question 290. Let H = span({[

11

]}. What is H⊥?

4.3 QR Decompositions

~ei is a great basis because it is orthogonal, normal, and a basis. How to dothis in general. Walk through creating an orthogonal basis from a basis inRn in P

Nicholas Long

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