Row Reduction & Echelon Forms - University of Connecticuttroby/Math2210S14/H8dd3n/cb.pdfRow...

1.2

Row Reduction & Echelon Forms

True or False: The reduced echelon form of a matrix is unique.

Tom Roby (UConn) Math 2210Q: Clicker Questions 22 January 2014 1 / 64

1.2


True or False: If every column of an augmented matrix contains apivot, then the corresponding system is consistent.


1.2


True or False: The pivot positions in a matrix depend on whetherinterchanges are used in the row reduction process.


1.2


True or False: A general solution of a system is an explicitdescription of all solutions of the system.


1.2


True or False: Whenever a system has free variables, the solutionset contains more than one solution.


1.3

Vector Equations

True or False: Asking whether the linear system corresponding toan augmented matrix [a1 a2 a3 b] has a solution amounts to askingwhether b is in Span{a1, a2, a3}.


1.3

Vector Equations

True or False: Any ordered list of five real numbers is a vector in R5.


1.4

Matrix Equations

True or False: If the equation Ax = b is consistent, then b is in theset spanned by the columns of A.


1.4

Matrix Equations

True or False: Any linear combination of vectors can always bewritten in the form Ax for a suitable matrix A and vector x.


1.4

Matrix Equations

True or False: If the coefficient matrix A has a pivot position inevery row, then the equation Ax = b is inconsistent.


1.4

Matrix Equations

True or False: If A is an m × n matrix whose columns do not spanRn, then the equation Ax = b is consistent for every b in Rn.


1.5

Solution Sets of Linear Equations

True or False: A homogenous system of equations can beinconsistent.


1.5


True or False: If x is a nontrivial solution of Ax = 0, then everyentry of x is nonzero.


1.5


True or False: A equation Ax = b is homogenous if the zero vectoris a solution.


1.5


True or False: If Ax = b is consistent, then the solution set ofAx = b is obtained by translating the solution set of Ax = 0.


1.7

Linear Independence

True or False: If u and v are linearly independent and w is inSpan{u, v}, then {u, v,w} is linearly dependent.


1.7

Linear Independence

True or False: If a set contains fewer vectors than there are entriesin the vectors, then the set is linearly independent.


1.7

Linear Independence

True or False: If a set in Rn is linearly dependent, then the setcontains more than n vectors.


1.8

Linear Transformations

True or False: The range of the transformation x 7→ Ax is the set ofall linear combinations of the rows of A.


1.8


True or False: Every matrix transformation is a lineartransformation.


1.8


True or False: A linear transformation preserves the operations ofvector addition and scalar multiplication.


1.8


True or False: A linear transformation T : Rn → Rm always mapsthe zero vector in Rn to the zero vector in Rm.


1.9

Matrix of a Linear Transformation

True or False: If A is a 4× 3 matrix, then the transformationx 7→ Ax maps R3 onto R4.


1.9


True or False: The columns of the standard matrix for a lineartransformation T from Rn to Rm are the images of the columns ofthe n × n identity matrix under T .


1.9


True or False: A mapping T : Rn → Rm is one-to-one if eachvector in Rn maps to a unique vector in Rm.


2.1

Matrix Operations

True or False: The first row of AB is the first row of A multipliedon the right by B .


2.1

Matrix Operations

True or False: If A and B are 3× 3 matrices andB = [b1 b2 b3 ], then

AB = [Ab1 + Ab2 + Ab3 ]


2.1

Matrix Operations

True or False: For any square matrix A, (A2)T = (AT )2.


2.1

Matrix Operations

True or False: The transpose of the sum of two matrices equals thesum of their transposes.


2.2

Matrix Inverses

True or False: If A is invertible, then any sequence of elementaryrow operations that transforms A to the identity matrix I alsotransforms A−1 to I .


2.2

Matrix Inverses

True or False: Any product of n × n invertible matrices is itselfinvertible, and its inverse is the product of its factors’ inverses in thesame order.


2.2

Matrix Inverses

True or False: If A is an n × n matrix and Ax = ej is consistent forevery j ∈ {1, 2, . . . , n}, then A is invertible. (Here ej denotes thestandard basis vector, i.e., the jth column of the identity matrix.)


2.2

Matrix Inverses

True or False: If A can be row-reduced to the identity matrix, thenA must be invertible.


2.3

Characterizing Matrix Inverses

True or False: Let A be an n× n matrix. If there is an n× n matrixD such that AD = I , then DA = I .


2.3


True or False: Let A be an n × n matrix. If the lineartransformation x 7→ Ax maps Rn into Rn, then the row-reducedechelon form of A is I .


2.3


True or False: Let A be an n × n matrix. If the columns of A arelinearly independent, then the columns of A span Rn.


2.3


True or False: Let A be an n × n matrix. If the equation Ax = bhas at least one solution for every b ∈ Rn, then the transformationx 7→ Ax is not one-to-one.


2.3


True or False: Let A be an n × n matrix. If there is a b ∈ Rn suchthat the equation Ax = b is consistent, then the solution is unique.


2.4

Partitioned (Block) Matrices

True or False: Let A be an k × ` matrix, and as usual let In denotethe n × n identity matrix. Then the matrix[

Ik A0 I`

]is invertible.


2.4

Partitioned (Block) Matrices

True or False: The columns of the following matrix span R51 0 0 1 20 1 0 3 40 0 1 1 20 0 0 1 00 0 0 0 1


3.1–2

Determinants

True or False: If A is an invertible matrix, then detA is the productof its diagonal entries.


3.1–2

Determinants

True or False: The cofactor expansion of detA down a column isthe negative of the cofactor expansion along the corresponding row.


3.1–2

Determinants

True or False: For any n × n matrix A

detAT = (−1) detA .


3.1–2

Determinants

True or False: If two successive column interchanges are made toan n × n matrix A, then the determinant remains unchanged.


3.1–2

Determinants

True or False: If detA = 0, then either a row or column is allzeroes, or there are two rows (or two columns) which are the same.


Ch. 3

Determinants

True or False: If A is a 2× 2 matrix with zero determinant, thenone column of A is a multiple of the other.


Ch. 3

Determinants

True or False: For any n × n matrix A

(detA)(detA−1) = 1


Ch. 4

Vector Spaces

True or False: A vector space is also a subspace.


Ch. 4

Vector Spaces

True or False: R2 is a subspace of R3.


Ch. 4

Vector Spaces

True or False: A subset H of a vector space V is a subsapce of V ifthe following conditions are satisfied:

1 the zero vector of V is in H ;

2 u, v, and u+v are in H ; and

3 c is a scalar, and cu is in H .


Ch. 4

Vector Spaces

True or False: The range of a linear transformation is a vectorspace.


Ch. 4

Vector Spaces

True or False: NulA is the kernal of the mapping x 7→ Ax. space.


Ch. 4

Vector Spaces

True or False: A linearly independent set in a subspace H is a basisfor H .


Ch. 4

Vector Spaces

True or False: If a finite set S of nonzero vectors spans a vectorspace V , then some subset of S is a basis for V .


Ch. 4

Vector Spaces

True or False: A basis is a linearly independent set that is as largeas possible.


Ch. 4

Vector Spaces

True or False: If B is an echelon form of a matrix A, then the pivotcolumns of B form a basis for ColA.


Ch. 4

Vector Spaces

True or False: If B is the standard basis for Rn, then theB-coordinate vector of some x ∈ Rn is x itself.


Ch. 4

Vector Spaces

True or False: The correspondence [x]B 7→ x is called thecoordinate mapping.


Ch. 4

Vector Spaces

True or False: A plane in R3 is always isomorphic to R2.


Ch. 5

Eigenvalues & Eigenvectors

True or False: If Ax = λx for some scalar λ, then x is aneigenvector of A.


Ch. 5


True or False: If v1 and v2 are linearly independent eigenvectors,then they correspond to distinct eigenvalues.


Ch. 5


True or False: A steady-state vector for a stochastic matrix isactually an eigenvector.


Ch. 5


True or False: An eigenspace of A is a null space of a certain matrix.


Ch. 5


True or False: A row replacement operation on A does not changethe eigenvalues.


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