MA 106 Linear Algebra

Sudhir R. Ghorpade

January 5, 2016

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Generalities about the Course

INSTRUCTORS: Prof. Santanu Dey (D1 & D3) andProf. Sudhir R. Ghorpade (D2 & D4)

LECTURES: D2: Mon 9.30, Tue 10.30, Thu 11.30.D4: Mon, Thu 3.30 - 5 pm, all in LA 102TUTS: Wed 3 - 4 pm in LT 004, 005, 006, 105, 106(for D2) and LT 304, 305, 306, 205, 206 (for D4)OFFICE: 106 B, First Floor, Maths Dept.OFFICE HOURS: Tue, 12 - 1 pm or by appointment.ATTENDANCE: Compulsory! Random name calling ineach class. We may not rely on biometric attendance.EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up)+ 1 Common Quiz + Final. Marks: 5 + 15 + 30 = 50.BONUS: 2.5 Extra Marks for 100 % Attendance;However, −1 mark for each absentee.MORE INFO: See the Moodle page of the course orwww.math.iitb.ac.in/∼srg/106

2/44

Generalities about the Course

INSTRUCTORS: Prof. Santanu Dey (D1 & D3) andProf. Sudhir R. Ghorpade (D2 & D4)LECTURES: D2: Mon 9.30, Tue 10.30, Thu 11.30.D4: Mon, Thu 3.30 - 5 pm, all in LA 102TUTS: Wed 3 - 4 pm in LT 004, 005, 006, 105, 106(for D2) and LT 304, 305, 306, 205, 206 (for D4)

OFFICE: 106 B, First Floor, Maths Dept.OFFICE HOURS: Tue, 12 - 1 pm or by appointment.ATTENDANCE: Compulsory! Random name calling ineach class. We may not rely on biometric attendance.EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up)+ 1 Common Quiz + Final. Marks: 5 + 15 + 30 = 50.BONUS: 2.5 Extra Marks for 100 % Attendance;However, −1 mark for each absentee.MORE INFO: See the Moodle page of the course orwww.math.iitb.ac.in/∼srg/106

2/44

Generalities about the Course

INSTRUCTORS: Prof. Santanu Dey (D1 & D3) andProf. Sudhir R. Ghorpade (D2 & D4)LECTURES: D2: Mon 9.30, Tue 10.30, Thu 11.30.D4: Mon, Thu 3.30 - 5 pm, all in LA 102TUTS: Wed 3 - 4 pm in LT 004, 005, 006, 105, 106(for D2) and LT 304, 305, 306, 205, 206 (for D4)OFFICE: 106 B, First Floor, Maths Dept.OFFICE HOURS: Tue, 12 - 1 pm or by appointment.

ATTENDANCE: Compulsory! Random name calling ineach class. We may not rely on biometric attendance.EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up)+ 1 Common Quiz + Final. Marks: 5 + 15 + 30 = 50.BONUS: 2.5 Extra Marks for 100 % Attendance;However, −1 mark for each absentee.MORE INFO: See the Moodle page of the course orwww.math.iitb.ac.in/∼srg/106

2/44

Generalities about the Course

INSTRUCTORS: Prof. Santanu Dey (D1 & D3) andProf. Sudhir R. Ghorpade (D2 & D4)LECTURES: D2: Mon 9.30, Tue 10.30, Thu 11.30.D4: Mon, Thu 3.30 - 5 pm, all in LA 102TUTS: Wed 3 - 4 pm in LT 004, 005, 006, 105, 106(for D2) and LT 304, 305, 306, 205, 206 (for D4)OFFICE: 106 B, First Floor, Maths Dept.OFFICE HOURS: Tue, 12 - 1 pm or by appointment.ATTENDANCE: Compulsory! Random name calling ineach class. We may not rely on biometric attendance.EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up)+ 1 Common Quiz + Final. Marks: 5 + 15 + 30 = 50.

BONUS: 2.5 Extra Marks for 100 % Attendance;However, −1 mark for each absentee.MORE INFO: See the Moodle page of the course orwww.math.iitb.ac.in/∼srg/106

2/44

Generalities about the Course

INSTRUCTORS: Prof. Santanu Dey (D1 & D3) andProf. Sudhir R. Ghorpade (D2 & D4)LECTURES: D2: Mon 9.30, Tue 10.30, Thu 11.30.D4: Mon, Thu 3.30 - 5 pm, all in LA 102TUTS: Wed 3 - 4 pm in LT 004, 005, 006, 105, 106(for D2) and LT 304, 305, 306, 205, 206 (for D4)OFFICE: 106 B, First Floor, Maths Dept.OFFICE HOURS: Tue, 12 - 1 pm or by appointment.ATTENDANCE: Compulsory! Random name calling ineach class. We may not rely on biometric attendance.EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up)+ 1 Common Quiz + Final. Marks: 5 + 15 + 30 = 50.BONUS: 2.5 Extra Marks for 100 % Attendance;However, −1 mark for each absentee.

MORE INFO: See the Moodle page of the course orwww.math.iitb.ac.in/∼srg/106

2/44

Generalities about the Course

INSTRUCTORS: Prof. Santanu Dey (D1 & D3) andProf. Sudhir R. Ghorpade (D2 & D4)LECTURES: D2: Mon 9.30, Tue 10.30, Thu 11.30.D4: Mon, Thu 3.30 - 5 pm, all in LA 102TUTS: Wed 3 - 4 pm in LT 004, 005, 006, 105, 106(for D2) and LT 304, 305, 306, 205, 206 (for D4)OFFICE: 106 B, First Floor, Maths Dept.OFFICE HOURS: Tue, 12 - 1 pm or by appointment.ATTENDANCE: Compulsory! Random name calling ineach class. We may not rely on biometric attendance.EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up)+ 1 Common Quiz + Final. Marks: 5 + 15 + 30 = 50.BONUS: 2.5 Extra Marks for 100 % Attendance;However, −1 mark for each absentee.MORE INFO: See the Moodle page of the course orwww.math.iitb.ac.in/∼srg/106

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What is Linear Algebra

WIKIPEDIA DESCRIPTION: Linear algebra is thebranch of mathematics concerning vector spaces andlinear mappings between such spaces. It includes thestudy of lines, planes, and subspaces, but is alsoconcerned with properties common to all vectorspaces.

MOREOVER: Linear algebra is classically related tothe study of:

Systems of linear equations and their solutionsMatricesDeterminants...

For a more specific decription, at least as far as thiscourse is concerned, see the official syllabus.

3/44

What is Linear Algebra

WIKIPEDIA DESCRIPTION: Linear algebra is thebranch of mathematics concerning vector spaces andlinear mappings between such spaces. It includes thestudy of lines, planes, and subspaces, but is alsoconcerned with properties common to all vectorspaces.MOREOVER: Linear algebra is classically related tothe study of:

Systems of linear equations and their solutionsMatricesDeterminants...

For a more specific decription, at least as far as thiscourse is concerned, see the official syllabus.

3/44

What is Linear Algebra

WIKIPEDIA DESCRIPTION: Linear algebra is thebranch of mathematics concerning vector spaces andlinear mappings between such spaces. It includes thestudy of lines, planes, and subspaces, but is alsoconcerned with properties common to all vectorspaces.MOREOVER: Linear algebra is classically related tothe study of:

Systems of linear equations and their solutionsMatricesDeterminants...

For a more specific decription, at least as far as thiscourse is concerned, see the official syllabus.

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Why Study Linear AlgebraSHORT ANSWER: Because it is beautiful!

BUT ALSO BECAUSE IT IS:One of the most important basic areas in all ofMathematics, having an impact comparable to that ofCalculus.Provides a vital arena where the interaction ofMathematics and machine computation is seen.Many of the problems studied in Linear Algebra areamenable to systematic and even algorithmicsolutions, and this makes them implementable oncomputers.Many geometric topics are studied making use ofconcepts from Linear Algebra.Applications to Physics, Engineering, Probability &Statistics, Economics and Biology.

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Why Study Linear AlgebraSHORT ANSWER: Because it is beautiful!BUT ALSO BECAUSE IT IS:

One of the most important basic areas in all ofMathematics, having an impact comparable to that ofCalculus.Provides a vital arena where the interaction ofMathematics and machine computation is seen.Many of the problems studied in Linear Algebra areamenable to systematic and even algorithmicsolutions, and this makes them implementable oncomputers.Many geometric topics are studied making use ofconcepts from Linear Algebra.Applications to Physics, Engineering, Probability &Statistics, Economics and Biology.

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Cartesian coordinate space

R denotes the set of all real numbers

The space Rn is the totality of all ordered n-tuples(x1, . . . , xn) where x1, . . . , xn vary over R. It is calledthe n-dimensional Euclidean space or then-dimensional Cartesian coordinate spaceElements of Rn are referred to as vectors whenn > 1. Elements of R may be referred to as scalars.For i = 1, . . . ,n, the function πi : Rn → R defined by

πi((x1, . . . , xn)) = xi

is called the i th coordinate function or the i th

coordinate projection.Given a function f : A→ Rn and 1 ≤ i ≤ n, thefunction fi : A→ R defined by fi := πi ◦ f is called thei th component function of f . These fi completelydetermine f . And we may write f = (f1, . . . , fn).

5/44

Cartesian coordinate space

R denotes the set of all real numbersThe space Rn is the totality of all ordered n-tuples(x1, . . . , xn) where x1, . . . , xn vary over R. It is calledthe n-dimensional Euclidean space or then-dimensional Cartesian coordinate space

Elements of Rn are referred to as vectors whenn > 1. Elements of R may be referred to as scalars.For i = 1, . . . ,n, the function πi : Rn → R defined by

πi((x1, . . . , xn)) = xi

is called the i th coordinate function or the i th

coordinate projection.Given a function f : A→ Rn and 1 ≤ i ≤ n, thefunction fi : A→ R defined by fi := πi ◦ f is called thei th component function of f . These fi completelydetermine f . And we may write f = (f1, . . . , fn).

5/44

Cartesian coordinate space

R denotes the set of all real numbersThe space Rn is the totality of all ordered n-tuples(x1, . . . , xn) where x1, . . . , xn vary over R. It is calledthe n-dimensional Euclidean space or then-dimensional Cartesian coordinate spaceElements of Rn are referred to as vectors whenn > 1. Elements of R may be referred to as scalars.

For i = 1, . . . ,n, the function πi : Rn → R defined by

πi((x1, . . . , xn)) = xi

is called the i th coordinate function or the i th

coordinate projection.Given a function f : A→ Rn and 1 ≤ i ≤ n, thefunction fi : A→ R defined by fi := πi ◦ f is called thei th component function of f . These fi completelydetermine f . And we may write f = (f1, . . . , fn).

5/44

Cartesian coordinate space

R denotes the set of all real numbersThe space Rn is the totality of all ordered n-tuples(x1, . . . , xn) where x1, . . . , xn vary over R. It is calledthe n-dimensional Euclidean space or then-dimensional Cartesian coordinate spaceElements of Rn are referred to as vectors whenn > 1. Elements of R may be referred to as scalars.For i = 1, . . . ,n, the function πi : Rn → R defined by

πi((x1, . . . , xn)) = xi

is called the i th coordinate function or the i th

coordinate projection.

Given a function f : A→ Rn and 1 ≤ i ≤ n, thefunction fi : A→ R defined by fi := πi ◦ f is called thei th component function of f . These fi completelydetermine f . And we may write f = (f1, . . . , fn).

5/44

Cartesian coordinate space

R denotes the set of all real numbersThe space Rn is the totality of all ordered n-tuples(x1, . . . , xn) where x1, . . . , xn vary over R. It is calledthe n-dimensional Euclidean space or then-dimensional Cartesian coordinate spaceElements of Rn are referred to as vectors whenn > 1. Elements of R may be referred to as scalars.For i = 1, . . . ,n, the function πi : Rn → R defined by

πi((x1, . . . , xn)) = xi

is called the i th coordinate function or the i th

coordinate projection.Given a function f : A→ Rn and 1 ≤ i ≤ n, thefunction fi : A→ R defined by fi := πi ◦ f is called thei th component function of f . These fi completelydetermine f . And we may write f = (f1, . . . , fn).

5/44

Algebraic structure of Rn

Addition: For x = (x1, . . . , xn),y = (y1, . . . , yn) define

x + y = (x1 + y1, . . . , xn + yn)

Usual laws of addition hold. We set:

0 = (0, . . . ,0), −x = (−x1, . . . ,−xn)

Scalar multiplication: For α ∈ R and x ∈ Rn, define

αx := (αx1, . . . , αxn).

The following properties clearly hold:Associativity: α(βx) = (αβ)x for all α, β ∈ R, x ∈ Rn

Distributivity: α(x + y) = αx + αy ∀ α ∈ R, x,y ∈ Rn

1x = x for all x ∈ Rn

6/44

Algebraic structure of Rn

Addition: For x = (x1, . . . , xn),y = (y1, . . . , yn) define

x + y = (x1 + y1, . . . , xn + yn)

Usual laws of addition hold. We set:

0 = (0, . . . ,0), −x = (−x1, . . . ,−xn)

Scalar multiplication: For α ∈ R and x ∈ Rn, define

αx := (αx1, . . . , αxn).

The following properties clearly hold:Associativity: α(βx) = (αβ)x for all α, β ∈ R, x ∈ Rn

Distributivity: α(x + y) = αx + αy ∀ α ∈ R, x,y ∈ Rn

1x = x for all x ∈ Rn

6/44

Algebraic structure of Rn

Addition: For x = (x1, . . . , xn),y = (y1, . . . , yn) define

x + y = (x1 + y1, . . . , xn + yn)

Usual laws of addition hold. We set:

0 = (0, . . . ,0), −x = (−x1, . . . ,−xn)

Scalar multiplication: For α ∈ R and x ∈ Rn, define

αx := (αx1, . . . , αxn).

The following properties clearly hold:Associativity: α(βx) = (αβ)x for all α, β ∈ R, x ∈ Rn

Distributivity: α(x + y) = αx + αy ∀ α ∈ R, x,y ∈ Rn

1x = x for all x ∈ Rn

6/44

Algebraic structure of Rn

Addition: For x = (x1, . . . , xn),y = (y1, . . . , yn) define

x + y = (x1 + y1, . . . , xn + yn)

Usual laws of addition hold. We set:

0 = (0, . . . ,0), −x = (−x1, . . . ,−xn)

Scalar multiplication: For α ∈ R and x ∈ Rn, define

αx := (αx1, . . . , αxn).

The following properties clearly hold:

Associativity: α(βx) = (αβ)x for all α, β ∈ R, x ∈ Rn

Distributivity: α(x + y) = αx + αy ∀ α ∈ R, x,y ∈ Rn

1x = x for all x ∈ Rn

6/44

Algebraic structure of Rn

Addition: For x = (x1, . . . , xn),y = (y1, . . . , yn) define

x + y = (x1 + y1, . . . , xn + yn)

Usual laws of addition hold. We set:

0 = (0, . . . ,0), −x = (−x1, . . . ,−xn)

Scalar multiplication: For α ∈ R and x ∈ Rn, define

αx := (αx1, . . . , αxn).

The following properties clearly hold:Associativity: α(βx) = (αβ)x for all α, β ∈ R, x ∈ Rn

Distributivity: α(x + y) = αx + αy ∀ α ∈ R, x,y ∈ Rn

1x = x for all x ∈ Rn

6/44

Algebraic structure of Rn

Addition: For x = (x1, . . . , xn),y = (y1, . . . , yn) define

x + y = (x1 + y1, . . . , xn + yn)

Usual laws of addition hold. We set:

0 = (0, . . . ,0), −x = (−x1, . . . ,−xn)

Scalar multiplication: For α ∈ R and x ∈ Rn, define

αx := (αx1, . . . , αxn).

The following properties clearly hold:Associativity: α(βx) = (αβ)x for all α, β ∈ R, x ∈ Rn

Distributivity: α(x + y) = αx + αy ∀ α ∈ R, x,y ∈ Rn

1x = x for all x ∈ Rn

6/44

Algebraic structure of Rn

Addition: For x = (x1, . . . , xn),y = (y1, . . . , yn) define

x + y = (x1 + y1, . . . , xn + yn)

Usual laws of addition hold. We set:

0 = (0, . . . ,0), −x = (−x1, . . . ,−xn)

Scalar multiplication: For α ∈ R and x ∈ Rn, define

αx := (αx1, . . . , αxn).

The following properties clearly hold:Associativity: α(βx) = (αβ)x for all α, β ∈ R, x ∈ Rn

Distributivity: α(x + y) = αx + αy ∀ α ∈ R, x,y ∈ Rn

1x = x for all x ∈ Rn

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2. LINEAR MAPS ON EUCLIDEAN SPACES

AND MATRICES

DefinitionA map f : Rn → Rm is said to be a linear if

f (αx + βy) = αf (x) + βf (y) ∀ α, β ∈ R, x,y ∈ Rn.

Examples:Projection map πi : Rn → R; inclusion mapRn → Rn+t ; multiplication by a fixed scalar

dot product by a fixed vector in Rn gives a linear mapfrom Rn to R; what about the converse?f : Rn → Rm linear⇔ fi linear for each i = 1, . . . ,m.Distance travelled is a linear function of time whenvelocity is constant. So is the voltage as a function ofresistance when the current is constant.|x |, xn (n > 1), sin x , etc. are not linear

7/44

2. LINEAR MAPS ON EUCLIDEAN SPACES

AND MATRICES

DefinitionA map f : Rn → Rm is said to be a linear if

f (αx + βy) = αf (x) + βf (y) ∀ α, β ∈ R, x,y ∈ Rn.

Examples:Projection map πi : Rn → R; inclusion mapRn → Rn+t ; multiplication by a fixed scalardot product by a fixed vector in Rn gives a linear mapfrom Rn to R; what about the converse?

f : Rn → Rm linear⇔ fi linear for each i = 1, . . . ,m.Distance travelled is a linear function of time whenvelocity is constant. So is the voltage as a function ofresistance when the current is constant.|x |, xn (n > 1), sin x , etc. are not linear

7/44

2. LINEAR MAPS ON EUCLIDEAN SPACES

AND MATRICES

DefinitionA map f : Rn → Rm is said to be a linear if

f (αx + βy) = αf (x) + βf (y) ∀ α, β ∈ R, x,y ∈ Rn.

Examples:Projection map πi : Rn → R; inclusion mapRn → Rn+t ; multiplication by a fixed scalardot product by a fixed vector in Rn gives a linear mapfrom Rn to R; what about the converse?f : Rn → Rm linear⇔ fi linear for each i = 1, . . . ,m.

Distance travelled is a linear function of time whenvelocity is constant. So is the voltage as a function ofresistance when the current is constant.|x |, xn (n > 1), sin x , etc. are not linear

7/44

2. LINEAR MAPS ON EUCLIDEAN SPACES

AND MATRICES

DefinitionA map f : Rn → Rm is said to be a linear if

f (αx + βy) = αf (x) + βf (y) ∀ α, β ∈ R, x,y ∈ Rn.

Examples:Projection map πi : Rn → R; inclusion mapRn → Rn+t ; multiplication by a fixed scalardot product by a fixed vector in Rn gives a linear mapfrom Rn to R; what about the converse?f : Rn → Rm linear⇔ fi linear for each i = 1, . . . ,m.Distance travelled is a linear function of time whenvelocity is constant. So is the voltage as a function ofresistance when the current is constant.

|x |, xn (n > 1), sin x , etc. are not linear

7/44

2. LINEAR MAPS ON EUCLIDEAN SPACES

AND MATRICES

DefinitionA map f : Rn → Rm is said to be a linear if

f (αx + βy) = αf (x) + βf (y) ∀ α, β ∈ R, x,y ∈ Rn.

Examples:Projection map πi : Rn → R; inclusion mapRn → Rn+t ; multiplication by a fixed scalardot product by a fixed vector in Rn gives a linear mapfrom Rn to R; what about the converse?f : Rn → Rm linear⇔ fi linear for each i = 1, . . . ,m.Distance travelled is a linear function of time whenvelocity is constant. So is the voltage as a function ofresistance when the current is constant.|x |, xn (n > 1), sin x , etc. are not linear

7/44

Exercise:

(i) Show that if f : Rn → Rm is a linear map then

f (k∑

i=1

αixi) =k∑

i=1

αi f (xi) ∀ xi ∈ Rn and αi ∈ R.

(ii) Show that the projection on a line L passing throughthe origin defines a linear map of R2 to R2 and itsimage is equal to L.

(iii) Show that rotation through a fixed angle θ is a linearmap from R2 → R2.

(iv) By a rigid motion of Rn we mean a map f : Rn → Rn

such that

d(f (x), f (y)) = d(x,y) ∀ x,y ∈ Rn.

Show that a rigid motion of R3 which fixes the originis a linear map.

8/44

Exercise:

(i) Show that if f : Rn → Rm is a linear map then

f (k∑

i=1

αixi) =k∑

i=1

αi f (xi) ∀ xi ∈ Rn and αi ∈ R.

(ii) Show that the projection on a line L passing throughthe origin defines a linear map of R2 to R2 and itsimage is equal to L.

(iii) Show that rotation through a fixed angle θ is a linearmap from R2 → R2.

(iv) By a rigid motion of Rn we mean a map f : Rn → Rn

such that

d(f (x), f (y)) = d(x,y) ∀ x,y ∈ Rn.

Show that a rigid motion of R3 which fixes the originis a linear map.

8/44

Exercise:

(i) Show that if f : Rn → Rm is a linear map then

f (k∑

i=1

αixi) =k∑

i=1

αi f (xi) ∀ xi ∈ Rn and αi ∈ R.

(ii) Show that the projection on a line L passing throughthe origin defines a linear map of R2 to R2 and itsimage is equal to L.

(iii) Show that rotation through a fixed angle θ is a linearmap from R2 → R2.

(iv) By a rigid motion of Rn we mean a map f : Rn → Rn

such that

d(f (x), f (y)) = d(x,y) ∀ x,y ∈ Rn.

Show that a rigid motion of R3 which fixes the originis a linear map.

8/44

Exercise:

(i) Show that if f : Rn → Rm is a linear map then

f (k∑

i=1

αixi) =k∑

i=1

αi f (xi) ∀ xi ∈ Rn and αi ∈ R.

(ii) Show that the projection on a line L passing throughthe origin defines a linear map of R2 to R2 and itsimage is equal to L.

(iii) Show that rotation through a fixed angle θ is a linearmap from R2 → R2.

(iv) By a rigid motion of Rn we mean a map f : Rn → Rn

such that

d(f (x), f (y)) = d(x,y) ∀ x,y ∈ Rn.

Show that a rigid motion of R3 which fixes the originis a linear map.

8/44

Exercise:

(i) Show that if f : Rn → Rm is a linear map then

f (k∑

i=1

αixi) =k∑

i=1

αi f (xi) ∀ xi ∈ Rn and αi ∈ R.

(ii) Show that the projection on a line L passing throughthe origin defines a linear map of R2 to R2 and itsimage is equal to L.

(iii) Show that rotation through a fixed angle θ is a linearmap from R2 → R2.

(iv) By a rigid motion of Rn we mean a map f : Rn → Rn

such that

d(f (x), f (y)) = d(x,y) ∀ x,y ∈ Rn.

Show that a rigid motion of R3 which fixes the originis a linear map.

8/44