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Matrix of a Linear
Mapping
4
...
2, we defined the standard matrix of a linear transformation L : IR
...
"' to be
the matrix whose columns are the images of the standard basis vectors S = { e1,
...
11 under L
...
We shall use the subscript S to distinguish the standard mat1ix of L:
11
The standard coordinates of the image under L of a vector 1 E IR
...
2)
These equations are exactly the same as those in Theorem 3
...
3 except that the notation
is fancier so that we can compare this standard description with a description with
respect to some other basis
...
2
...
11
...
,v11} be a basis for IR
...
-- IR
...
11
Then, for any 1 E IR
...
2) where the matrix
standard matrix of L
...
'v,,} is any basis for IR
...
11
--
IR
...
Define the matrix of the linear operator L with respect to the basis 13 to be
the matrix
We then have that for any 1 E IR
...
:B are the
...
3-basis vectors under L
...
3
...
EXAMPLE 1
Let L : IR
...
be defined by L(x1,x2,X3) = (x1 +2x2 - 2x3,-x2 +2x3,X1 + 2x2)
{[ :J [: l [j]}·
[H
�
=
...
=
Solution: By definition, the columns of [L]
...
3-coordinates of the images of
the vectors in
...
So, we find these images and write them as a linear combi
nation of the vectors in
...
-l
...
l
...
:B=
1
0
0
1
-2
][� ·
+ (-2)
+
[L]
...
:B
-2
+ (-2)
+
Hence,
=
+
1,
1)]
...
:B]
2 =
-2
5
]
[ -1 1
0
We can verify that this answer is correct by calculating L(x) in two ways
...
=
ilien
1=1
l :l l:J il-�l Ul
=
+2
+
=
EXAMPLE 1
[ �]
and by definition of the mapping, we have L(x)
(continued)
[L(x)]!B
=
L(4, 1, -2)
=
=
(10, -5, 6)
...
then
-11
EXAMPLE2
Let v
=
[! ]
...
2
...
IR2 was found to be
[pWJ ;t ]S
[
=
9/25
12/25
12/25
16/25
:
JR2
�
]
Find the matrix of projv with respect to a basis that shows the geometry of the trans
formation more clearly
...
Then, with 13
projv v
=
projv
proj" w
=
proj"
=
1 v +Ow
=
=
=
E
{V, w}, by geometry,
[!] [!]
[-�J [�]
and [projv w]!B
We now consider projv x for any x
=
rnl
=
Ov +Ow
Thus,
JR2
...
adapted to the geometry of the transformation (see Figure 4
...
4 ) This example will be
discussed further below
...
6
...
Of course, we want to generalize this to any linear operator L
:
V - V on a vector
space V with respect to any basis '13 for V
...
Let '13 = {v1,
...
Then, for any x EV, we can write x = biv1 +
·
·
·
+ b11v11• Therefore,
Taking '13-coordinates of both sides gives
[L(x)]21 = [b1L(v1) +
·
·
= b1 [L(vi)]21+
[
= [L(v1)l•
·
+ b11L(v11)]21
·
·
·
·
·
·
+ bn[L(v,,)]21
[L(v,)]•
f]
We make the following definition
...
, v11} is any basis for a vector space V and that L: V - V is
Matrix of a Linear Operator
a linear operator
...
EXAMPLE3
Solution: We have
(continued)
L(l)=l+x2
L(x) =1 +x +x2
L(x2) =x2
Hence,
a+b
We can check our answer by observing that [L(a+bx+cx2)]1l =
b
and
a+b+c
[
O
l
a
a+b
b =
b
c
a+b+c
[ l[ ] [
1 1
L
[ (a +bx+cx2)]1l = [L]1l[a+bx+ cx2]1l = 0 1 0
1 1 1
EXAMPLE4
Let L: P
l
P be defined by L(a+bx+cx2) =(a+b)+bx+(a+b+c)x2
...
�
Solution: We have
L(l +x+x2) =2 +x +3x2= 3
( )(1 - x)+(-3/2)(2)
( )1
( +x +x2)+ 2
L(l - x) =0 +(-l)x +Ox2 = 0
( )1
( +x +x2)+ (1)(1- x)+ (-1/2)(2)
L(2)= 2 +2x2= 2
( )1
( - x)+ (-1)(2)
( )1
( +x+x2)+ 2
Hence,
EXERCISE 1
Let L
:
([ ]) [
...
M2
( ,2) � M2
( ,2) be defined by L
matrixofLwith respect to the basis'B=
a
c
b
d
a+b
=
c
a-b
a+b+d
...
To make this as general as possible,
we would like to define the matrixcL
[ 1l
] of a linear mappingL: V
�
W, where '13 is
a basis for the vector space V and C is a basis for the vector space W
...
Change of Coordinates and Linear Mappings
In Example 2, we used special geometrical properties of the linear transformation and
of the chosen basis 13 to determine the 13-coordinate vectors that make up the 13-matrix
[L]s of the linear transformation L
:
2
2
IR
...
• In some applications of these ideas,
the geometry does not provide such a simple way of determining [Ls
]
...
11 - IR
...
Let L : IR
...
11 be a linear operator
...
11 and
let 13 be any other basis for IR
...
11),
we get
[L(x)s
J
P-1 [L(x)s
J
=
Substitute for [L(x)]s and [L(x)]s to get
But, P[x]s
=
[xs
] , so we have
Since this is true for every [x]s, we get, by Theorem 3
...
4, that
[Ls
]
=
p-I [L]sP
Thus, we now have a method of determining the 13-matrix of L, given the standard
matrix of Land a new basis 13
...
EXAMPLES
Let v
=
[ !]
...
Let us verify that the change of basis method just described
does transform the standard matrix [projil ]s to the 13-matrix [projil ]s
...
Hence, the
3125
-
...
The inverse is found to be P-1
...
by
-mat
...
v 1s
3/25
-4/25
4/25
3/25
][
9/25
1 2/2 5
Thus, we obtain exactly the same 13-matrix [projvs
] as we obtained by the earlier
geometric argument
...
Method 1
...
3
Hence,
=
=
s
[proJv 1]3
=
[proJi13[x3
]
]
Method 2
...
[proJv x]s
Therefore,
[projv]3[x3
...
[1 OJ [-14/25
1
p
=
=
under projil
...
[25] [129/25
12/25]
[5]
[69/25
]
/25 16/25 2 92/25
[-4/25
3/25 43/25
/25] [6992/25
/25] [23/25]
[proJ v ]s
...
W hat is really important is that it is easy to get a geometrical
understanding of what happens to vectors if you multiply by
[� �]
(the B-matrix); it
is much more difficult to understand what happens if you multiply by
[ 1;;�; ��;�;]
(the standard matrix)
...
EXAMPLE6
Let Lbe the linear mapping with standard matrix A
=
[{ �] , [-�]}
[; n
[� -n
Let B
=
be
a basis for IR
...
Find the matrix of L with respect to the basis B
...
_
=
p-1AP
=
�[ 1
4 -1
13] [24
=
-1] [13/2 1/2]
1 21/2 1/2
35] [31
=
EXAMPLE 7
Let Lbe the linear mapping with standard matrix A
=
=
It follows that the B-matrix of Lis
[L]3
'B
and we have p-1
{[i], m, [:]}
=
[
-3
7
-7
59
7
-5
-5-3
1
...
What does this mean in terms of the
geometry of the linear transformation? From the definition of [L]23 and the definition
of 13-coordinates of a vector, we see that the linear transformation stretches the first
basis vector
vector
vector
[:]
[:1
[i]
by a factor of 2, it reflects (because of the minus sign)the second basis
in the origin and stretches it by a factor of 3, and it stretches the third basis
by a factor of 4
...
This picture is not obvious from looking at the standard
matrix A
...
However, in order to deal
with these questions, one more computational tool is needed, the determinant, which
is discussed in Chapter 5
...
6
Practice Problems
Al Determine the matrix of the linear mappingL with
respect to the basis 13 in the following cases
...
2
In
JR
, 13 {v1,v2} andL(v1) v2,
a
( )
=
L(v2)
=
2v1 - v2; [ xJ23
=
[�]
=
=
=
[ X J
...
In each
of the following cases, assume that L is a linear
mapping and determine [L]23
...
[
-!
}
]
]
ml n
=
A3 Consider the basis 13
=
{v 1, v2,v3}
=
3
of R
...
Determine [L(v\)]2l, [L(v2)]2l, and [L(v3)]2l and
hence determine [Lk
[( :JJ nl· ([ �]] = [:]
...
(a) refic1,-2)
(b) projc2,1,-1)
(c) refic-1,-1,1)
AS (a) Hnd the coordinates of
m
with respect to the
ml·Hl·[!lJinR'
...
L(l,
[j]
[
=
{[j]
...
3 � JR
...
(c) Use parts (a) and (b) to determine
2, 4)
...
3
formation such that
·
�
R'
...
3 is a linear trans
0) = 0,
A7 Assume that each of the following matrices is the
standard matrix of a linear mappingL : JR
...
n
...
You may find it helpful to use a computer
to find inverses and to multiply matrices
...
[
13=
[
]
{
;
�]
...
]}�
[� - 6 ]
...
[� -� Hs=mrnrnl}
! s=
l � �tH mrnrn11
A8 Find the 13-matrix of each of the following linear
mappings
...
3 � JR
...
m
...
2
(b) L : P2 � P2 defined by L(a+ bx+ cx )
2
2
2
a+ (b+ c )x , 13
{ + x , + x, 1 x+ x }
2
(c) D : P2 � P2 defined by D(a+ bx+ cx )
2
b+
13
x, x }
(d) T : U � U, where U is the subspace of
upper-triangular matrices in
defined
=
by
T
-
a
c
...
Deter
mine [L(1)]2l for the given [ 1)$,
2
111 + 3112,
{V1, 112} and L(111)
(a) In JR
...
Determine the 13-matrix ofL
...
=
= Hl
=2
111 3v2,
{V1,112,v3}andL(111)
(b) InlR
...
B2 Consider the basis
13= { [�],[
_
�]}
of JR2
...
:a
...
:a, [L (v2)]
...
:a
...
and hence
and
[L (v3)]
...
13
(as in Ex
and determine the 13-matrix of the
L (-1,0,1)
(-2,0, 2)
...
L (l,4,4)
...
3
[�1 6 n 13= {[!] ' [�] }
[� �160l 13= {[ ] ' [ �]}
; s=
lO
...
basis
You may find it helpful to use a computer
to find inverses and to multiply matrices
...
Determine the matrix of
13
L
x
...
:a
with respect to the given
and use it to determine
given vector
for the
You may find it helpful to use a
computer to find inverses and to multiply matrices
...
formation such that
basis
(a) perp( ,2)
3
(b) perpc2,1,-2)
(c) reflc1,2, )
3
(b) Suppose that
3
of R3• In each of the following
cases, assume that
(a)
basisS
=
with respect to the
(b) Suppose that
(a)
with respect to the
inR3
·
·
JR3
-
...
L
...
:
transformation such that
and
Determine the 13-matrix of
(c) Use parts (a) and (b) to determine
(b)
(c)
[-�� �1�]
...
[xJ
...
:a
[
{
r3 6 =�l [�J
...
1 = r-n
13
=
U =�� =�n
[
XJ• [il
=
s=
}
{[�] [-;]
...
•
(d)
{
li =: H Ul r l lW
nl
�=
(c) L: M(2,2)
·
B9 Find the $-matrix of each of the following linear
�
3
IR
...
[� �]}
b
L
[XJ
...
3
(a) L : IR
...
Conceptual Problems
Dl Suppose that
13
and C are bases for IR
...
11• Suppose that P is the change
of coordinates matrix from
13
to S and that Q is
the change of coordinates matrix from C to S
...
2
13=
2
b + 2cx,
a1x2,
ordinates matrix from
x + x2}
to the standard basis
...
? (Hint: Consider the equation
2
AP = PD, or A v1
[
[�1 � ] =
]
[
v = v1
2
]
v D
...
,v },
11
let W be a vector space with basis C , and let
D3 Let V be a vector space with basis
L: V
�
W be a linear mapping
...
2
and C
...
2 and let P denote the change of co
13
IR
...
Let
13
pings with respect to the given bases
(a) D : P
�
and C
...
11 � IR
...
Express the
matrix [L]c in terms of [L]21, P, and Q
...
2
13
�
�
M(2,2) defined by
b
...
IR
...
13
c= {[�]·[�]}
= {1 + x2, 1 + x, - 1 + x + x2},
=