3.V.1. Changing Representations of Vectors3.V.2. Changing Map Representations

3.V. Change of Basis

3.V.1. Changing Representations of Vectors

Definition 1.1: Change of Basis MatrixThe change of basis matrix for bases , V is the representation of the identity map id : V → V w.r.t. those bases.

1 nid S β βB D D D dimV n

Lemma 1.2: Changing Basis

id

v vD BB D V v

Proof:

1

1 n

n

v

id

v

v β βB

BB D D D

B B

1

n

k kk

v

βB

D

vD 1

n

k kk

v

βB D

Alternatively,

id id

v vBB D DvD

Example 1.3:

2 1,

1 0

B

1 1,

1 1

D

1

23

2

D

1

21

2

D

1

21

2

D

21

2

1id

β E DD

22

1

0id

β E DD

1 1

2 23 1

2 2

B D

i 1 2 id β βB D D D

2

2

1 1

1 1id

D E

D E

2

1

1 1 2

1 1 1

D E

2

2

1

2

2 1 0

1 0 1id

eE B

B E

1

2

B

1 11 12 22 3 1 2

2 2

id

B D

B B

B D

i

2

2

1 11 12 2

1 1 1 1

2 2

id

D E

D E

D D

2

0

1

e

→

Lemma 1.4: A matrix changes bases iff it is nonsingular.

Proof : Bases changing matrix must be invertible, hence nonsingular.

Proof : (See Hefferon, p.239.)

Nonsingular matrix is row equivalent to I.

Hence, it equals to the product of elementary matrices, which can be shown to represent change of bases.

Corollary 1.5:A matrix is nonsingular it represents the identity map w.r.t. some pair of bases.

Exercises 3.V.1.

1. Find the change of basis matrix for , 2.

(a) = 2 , = e2 , e1 (b) = 2 , 1 1

,2 4

D

1 1,

2 4

B = 2 (c

)

1 2,

1 2

B(d

)

0 1,

4 3

D

2. Let p be a polynomial in 3 with

0

1Rep

1

2

p

B

B

where = 1+x, 1x, x2+x3, x2x3 . Find a basis such that

1

0Rep

2

0

p

D

D

3.V.2. Changing Map Representations

ˆ ˆˆ h

H

B D ˆ ˆid id

HD D B B

Example 2.1: Rotation by π/6 in x-y plane t : 2 → 2

2 2t T E E

cos sin6 6

sin cos6 6

3 1

2 2

1 3

2 2

Let1 0ˆ ,1 2

B

1 2ˆ ,0 3

D

2 2ˆ ˆ ˆ ˆ

ˆ t id id

T TB D E D B E

2

2

ˆˆ

1 0

1 2id

B EB E

2

2

1

ˆˆ

1 2

0 3id

E D

D E

2ˆ

21

31

03

E D

2

2 2 2

ˆ

ˆ

2 3 11 1 03 2 2ˆ

1 1 21 303 2 2

TB E

E D E E

ˆ ˆ

1 15 3 3 2 3

6 3

1 31 3

6 3

B D

Let 1

3

v →

3 112 231 3

2 2

w T v

13 3

21

1 3 32

2ˆ ˆid

v v

B E B

11 0 1

1 2 3

ˆ

1

1

B

ˆ

ˆ

ˆ ˆ

1 15 3 3 2 3

16 3ˆ11 3

1 36 3

T vB

B

B D

ˆ

111 3 3

61

1 3 36

D

2ˆid

w

E D

Example 2.2:

:

x y z

t y x z

z x y

→

3 3t T E E

0 1 1

1 0 1

1 1 0

1 0 0 0 1 1

0 , 1 , 0 1 , 0 , 1

0 0 1 1 1 0

t

∴

Let

1 1 1

1 , 1 , 1

0 2 1

B

Then

3

1 1 1

1 1 1

0 2 1

id

B E

3 3t id id TB B E B B E

1 0 0

0 1 0

0 0 2

Consider t : V → V with matrix representation T w.r.t. some basis.

If basis s.t. T = t → is diagonal,

Then t and T are said to be diagonalizable.

Definition 2.3: Matrix Equivalent

Same-sized matrices H and H are matrix equivalent

if nonsingular matrices P and Q s.t.

H = P H Q or H = P 1 H Q 1

Corollary 2.4:Matrix equivalent matrices represent the same map, w.r.t. appropriate pair of bases.

Matrix equivalence classes.

Elementary row operations can be represented by left-multiplication (H = P H ).

Elementary column operations can be represented by right-multiplication ( H = H Q ).

Matrix equivalent operations cantain both (H = P H Q ).

∴ row equivalent matrix equivalent

Example 2.5:

1 0

0 0

and1 1

0 0

are matrix equivalent but not row equivalent.

Theorem 2.6: Block Partial-Identity FormAny mn matrix of rank k is matrix equivalent to the mn matrix that is all zeros except that the first k diagonal entries are ones.

k k k n k

m nm k k m k n k

I OM

O O

Proof:

Gauss-Jordan reduction plus column reduction.

Example 2.7:1 2 1 1

0 0 1 1

2 4 2 2

A

G-J row reduction:

1 1 0 1 0 0 1 2 1 1 1 2 0 0

0 1 0 0 1 0 0 0 1 1 0 0 1 1

0 0 1 2 0 1 2 4 2 2 0 0 0 0

Column reduction:

1 2 0 0 1 0 0 01 2 0 0 1 0 0 0

0 1 0 0 0 1 0 00 0 1 1 0 0 1 0

0 0 1 0 0 0 1 10 0 0 0 0 0 0 0

0 0 0 1 0 0 0 1

Column swapping:

1 0 0 01 0 0 0 1 0 0 0

0 0 1 00 0 1 0 0 1 0 0

0 1 0 00 0 0 0 0 0 0 0

0 0 0 1

Combined:

1 0 2 01 1 0 1 2 1 1 1 0 0 0

0 0 1 00 1 0 0 0 1 1 0 1 0 0

0 1 0 12 0 1 2 4 2 2 0 0 0 0

0 0 0 1

Corollary 2.8: Matrix Equivalent and RankTwo same-sized matrices are matrix equivalent iff they have the same rank. That is, the matrix equivalence classes are characterized by rank.

Proof. Two same-sized matrices with the same rank are equivalent to the same block partial-identity matrix.

Example 2.9:The 22 matrices have only three possible ranks: 0, 1, or 2. Thus there are 3 matrix-equivalence classes.

If a linear map f : V n → W m is rank k,

then some bases → s.t. f acts like a projection n → m.

1 1

1 0

0

k k

k

n

c c

c c

c

c

DB

Exercises 3.V.2.

1. Show that, where A is a nonsingular square matrix, if P and Q are nonsingular square matrices such that PAQ = I then QP = A1 .

2. Are matrix equivalence classes closed under scalar multiplication? Addition?

3. (a) If two matrices are matrix-equivalent and invertible, must their inverses be matrix-equivalent?(b) If two matrices have matrix-equivalent inverses, must the two be matrix- equivalent?(c) If two matrices are square and matrix-equivalent, must their squares bematrix-equivalent?(d) If two matrices are square and have matrix-equivalent squares, must they be matrix-equivalent?