3.III.1. Representing Linear Maps with Matrices3.III.2. Any Matrix Represents a Linear Map

3.III. Computing Linear Maps

3.III.1. Representing Linear Maps with Matrices

A linear map is determined by its action on a basis.

Example 1.1:

Let 1 2

2 1, ,

0 4

β βB 1 2 3

1 0 1

, , 0 , 2 , 0

0 0 1

δ δ δD

12

10

1

11

24

0

h: 2 → 3 by

1 1 0 1

1 0 2 0

1 0 0 1

a b c

→

0

1

21

a

b

c

→ 1

0

1

21

h

β

D

1 1 0 1

2 0 2 0

0 0 0 1

a b c

→1

1

0

a

b

c

→ 2

1

1

0

h

β

D

Given 1 21 1 2 2

2

cc c

c

v β βB

R

1

2

ch h

c

v

B

1 2

01

11

201

c c

D

D

2

1 2

1

1

2

c

c c

c

D

1 1 2 2c h c h β β

E.g.

4 1

8 2

B

4 1

8 2h h

B

2

5

21

D

01

11 2 1

201

D

D

Matrix notation:

1 1 2 2h c h c h v β βD D

1

2

0 1

11

21 0

c

c

B

11 2

2

ch h

c

β β

D DB

, 1 2Rep h h h h β βB D B D D D

Definition 1.2: Matrix Representation

Let V and W be vector spaces of dimensions n and m with bases and . The matrix representation of linear map h: V → W w.r.t. and is an mn matrix

, 1Rep nh h h h β βB D B D D D

1k

k

mk

h

h

h

βD

D

where

Example 1.3:

h: 3 → 1 by 1

2 1 2 3

3

2

a

a a a a x

a

1 2 3

0 0 2

, , 0 , 2 , 0

1 0 0

β β βBLet 1 , 1x x D

1h xβ 2 2h β

3 4h β

1

21

2

D

1

1

D

2

2

D

11 2

21

1 22

h

B D

B D

Theorem 1.4: Matrix Representation

Let H = ( hi j ) be the matrix rep of linear map h: V n → W m w.r.t. bases and .

Then

h v H vB BD

11

1

n

k kk

n

mk kk

h c

h c

D

Proof: Straightforward (see Hefferon, p.198 )

where1

n

c

c

vB

B

Definition 1.5: Matrix-Vector ProductThe matrix-vector product of a mn matrix and a n1 vector is

1111 1 1

1

1

n

k kkn

nm mn n

mk kk

a ca a c

a a ca c

1

m

ρ c

ρ c

1

n

k kk

c

χ

Example 1.6: (Ex1.3) h: 3 → 1 by 1

2 1 2 3

3

2

a

a a a a x

a

1 2 3

0 0 2

, , 0 , 2 , 0

1 0 0

β β βB

1 , 1x x D

11 2

21

1 22

h

B D

B D

Task: Calculate where h sends4

1

0

v

0

1

22

vB

B

h h v vB B D BD

1 01 22 1/ 21

1 2 22

BB D

14

21

42

D

9

29

2

D

or

9 91 1

2 2x x 9

Example 1.7:

Let π: 3 → 2 be the projection onto the xy-plane.

And1 1 1

0 , 1 , 0

0 0 1

B2 1

,1 1

D

→ 1 1 1, ,

0 1 0

B

1 0 1, ,

1 1 1

D D D

Illustrating Theorem 1.4 using

1 0 1

1 1 1

B D

B D

2

2

v

1

2

1

v

B

1

1 0 12

1 1 11

vB DB D

B

2

2

1

v

0

2

D

2

2

→

2

1 0 02

0 1 01

v2

2

→1 0 0

0 1 0

3

1 0 0, ,

0 1 0

E

Example 1.8: Rotation

Let tθ : 2 → 2 be the rotation by angle θ in the xy-plane.

2

1 0,

0 1t t

Ecos sin

,sin cos

→

cos sin

sin cost

E.g. / 6

3 13 32 22 21 3

2 2

t

3.598

0.232

Example 1.10: Matrix-vector product as column sum2

1 0 1 1 0 11 2 1 1

2 0 3 2 0 31

1

7

Exercise 3.III.1.

Using the standard bases, find(a) the matrix representing this map;(b) a general formula for h(v).

1. Assume that h: 2 → 3 is determined by this action.

21

20

0

0

01

11

2. Let d/dx: 3 → 3 be the derivative transformation.(a) Represent d/dx with respect to , where = 1, x, x2, x3 .(b) Represent d/dx with respect to , where = 1, 2x, 3x2, 4x3 .

3.III.2. Any Matrix Represents a Linear Map

Theorem 2.1:Every matrix represents a homomorphism between vector spaces, of appropriate dimensions, with respect to any pair of bases.

Proof by construction:

Given an mn matrix H = ( hi j ), one can construct a homomorphism

h: V n → W m by v h(v ) with h(v ) = H · v

where and are any bases for V and W, resp.

v is an n1 column vector representing vV w.r.t. .

Example 2.2: Which map the matrix represents depends on which bases are used.

Let1 0

0 0

H 1 1

1 0,

0 1

B D 2 2

0 1,

1 0

B D

Then h1: 2 → 2 as represented by H w.r.t. 1 and 1 gives

1

1 1

2 2

c c

c c

B

1

0

c

While h2: 2 → 2 as represented by H w.r.t. 2 and 2 gives

1 1 1

1

2

1 0

0 0

c

c

B D B

1

1

0

c D

2

1 2

2 1

c c

c c

B 2

0

c

2 2 2

2

1

1 0

0 0

c

c

B D B

2

2

0

c D

Convention:

An mn matrix with no spaces or bases specified will be assumed to represent

h: V n → W m w.r.t. the standard bases.

In which case, column space of H = (h).

Theorem 2.3:rank H = rank h

Proof: (See Hefferon, p.207.)

For each set of bases for h: V n → W m , Isomorphism: W m → m.

∴ dim columnSpace = dim rangeSpace

Example 2.4: Any map represented by

1 2 2

1 2 1

0 0 3

0 0 2

H must be of type h: V 3 → W 4

rank H = 2 → dim (h) = 2

Corollary 2.5: Let h be a linear map represented by an mn matrix H. Then

h is onto rank H = m h is 1-1 rank H = n

Corollary 2.6:A square matrix represents nonsingular maps iff it is a nonsingular matrix.A matrix represents an isomorphism iff it is square and nonsingular.

Example 2.7:Any map from 2 to 1 represented w.r.t. any pair of bases by

1 2

0 3

H

is nonsingular because rank H = 2.

Example 2.8:Any map represented by

1 2

3 6

H is singular because H is singular.

Exercise 3.III.2.

1. Decide if each vector lies in the range of the map from 3 to 2 represented with respect to the standard bases by the matrix.

1 1 2 1,

0 1 4 3

(a)

2 0 3 1,

4 0 6 1

(b)

2. Describe geometrically the action on 2 of the map represented with respect to the standard bases 2 , 2 by this matrix.

3 0

0 2

Do the same for these:

1 0

0 0

0 1

1 0

1 3

0 1