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3.II.1. Representing Linear Maps with Matrices3.II.2. Any Matrix Represents a Linear Map
3.II. Computing Linear Maps
3.II.1. Representing Linear Maps with Matrices
A linear map is determined by its action on the bases.
Consider h : V n → W m with bases and for V and W , resp.
Range space of h = Range h = span h() = subspace of W.
Rank h = dim[Range h ] min( n, m )
Definition: Matrix Representation
The matrix representation of linear map h: V n → W m 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:
Let1 2, β βB
3 1 2 3
1 0 0
, , 0 , 1 , 0
0 0 1
e e eD E
12
10
1
11
24
0
h: 2 → 3 by
1
1
1
1
h
β 2
1
2
0
h
β
2 1,
0 4
1 0,
0 1
B B
Given 1 21 1 2 2
2
cc c
c
v β βB
R
1
2
ch h
c
v
B
1 1 2 2c h c h β β 3
11 2
2
ch h
c
β β
B EB
3
1
2
ch
c
B E
B
3 3 3
1 2 Rep h h h h
H β βB E B E B E
3
1 1
1 2
1 0
B E
1 1
1 , 2
1 0
Range h span
i j j i
h h β
Let 1 2
2 1, ,
0 4
β βB 1 2 3
1 0 1
, , 0 , 2 , 0
0 0 1
δ δ δD
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
0 1
1Rep 1
21 0
h h
HB D B D
B D
2
a c
b
c
→
2
a c
b
c
01
1, 1
201
Range h span
D
D
Column space H
E.g. 4 1
8 2
vB
1
2h
B D
2
5
21
D
01
11 2 1
201
D
D
0 1
111
221 0
B
B D
1
2h h
v
B3
1 11
1 22
1 0
B
B E
1 1
1 1 2 2
1 0
3
5
1
1 0 15
2 0 2 1 02
0 0 1
3
5
1
i kk
i kh v H v
1
1i i ni
n
v
h h
v
H v
Example: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
Task: Calculate where h sends4
1
0
v
0
1
22
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
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.II.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.II.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.
i j j ih h β 1 2
Rep h h h h H β βB D B D B D
h : V n → W m with bases and s.t.
Proof by construction:
Let H be an mn matrix. Then there is an homomorphism
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 All rows L.I. ( dim = dim W ) h is 1-1 rank H = n All cols L.I. ( dim = dim V )
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.
Definition:A matrix is nonsingular if it is square and the corresponding linear system has a unique solution.A linear map is nonsingular if it is one-to-one.
Exercise 3.II.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