3.v.1. changing representations of vectors 3.v.2. changing map representations 3.v. change of basis
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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?