Page 219 - 35Linear Algebra

11.2 Matrix of a Linear Transformation (Redux) 219

i
The number m is the ith component of L(e j ) in the basis F, while the f i
j
are vectors (note that if α is a scalar, and v a vector, αv = vα, we have
used the latter—rather uncommon—notation in the above formula). The
i
numbers m naturally form a matrix whose jth column is the column vector
j
displayed above. Indeed, if
n
1
v = e 1 v + · · · + e n v ,

Then

n
2
1
L(v) = L(v e 1 + v e 2 + · · · + v e n )
m
X
2
1
n
= v L(e 1 ) + v L(e 2 ) + · · · + v L(e n ) = L(e j )v j
j=1
" #
m n m
X X X
1
m
i j
j
= (f 1 m + · · · + f m m )v = f i M v
j j j
j=1 i=1 j=1
 1 1 1   
1
m 1 m 2 · · · m n v
v
 m 2 m 2   2
1 2
= f 1 f 2 · · · f m  . .   

.
 . . . . . .   .
.   .
m m · · · m m v n
1 n
In the column vector-basis notation this equality looks familiar:
     
v 1 m 1 . . . m 1 v 1
1 n
 .
.
.
 .  .   . 
L  .  =  . . . .   .  .
v n m m . . . m m v n
E 1 n F
i
The array of numbers M = (m ) is called the matrix of L in the input and
j
output bases E and F for V and W, respectively. This matrix will change
if we change either of the bases. Also observe that the columns of M are
computed by examining L acting on each basis vector in V expanded in the
basis vectors of W.
Example 123 Let L: P 1 (t) 7→ P 1 (t), such that L(a + bt) = (a + b)t. Since V =
219