1 V-22 Update for AHS - Changing the Conversation Team Osprey.
3.V.1. Changing Representations of Vectors 3.V.2. Changing Map Representations 3.V. Change of Basis.
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Transcript of 3.V.1. Changing Representations of Vectors 3.V.2. Changing Map Representations 3.V. Change of Basis.
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?