3.iii.1. representing linear maps with matrices 3.iii.2. any matrix represents a linear map 3.iii....
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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