18.5. THE MATRIX OF A LINEAR TRANSFORMATION 439
Thus interchanging the order in the sum on the left,
∑j
(∑
iai jwi
)b j = ∑
j(Lv j)b j,
and this must hold for all b which happens if and only if
Lv j = ∑i
ai jwi (18.8)
It may help to write 18.8 in the form(Lv1 · · · Lvn
)=(
w1 · · · wm
)A (18.9)
where [L] = A = (ai j) .A little less precisely, you need for b ∈ Fn,(
w1 · · · wm
)Ab = L
(v1 · · · vn
)b =
(Lv1 · · · Lvn
)b (18.10)
where we use the usual conventions for notation like the above. Then since 18.10 is to holdfor all b, 18.9 follows.
Example 18.5.3 Let
V ≡ { polynomials of degree 3 or less},
W ≡ { polynomials of degree 2 or less},
and L≡D where D is the differentiation operator. A basis for V is {1,x, x2, x3} and a basisfor W is {1,x, x2}.
What is the matrix of this linear transformation with respect to this basis? Using 18.9,(0 1 2x 3x2
)=(
1 x x2)[D] .
It follows from this that
[D] =
0 1 0 00 0 2 00 0 0 3
.
Now consider the important case where V = Fn, W = Fm, and the basis chosen is the
standard basis of vectors ei ≡(
0 · · · 1 · · · 0)T
the 1 in the ith position. Let L bea linear transformation from Fn to Fm and let [L] be the matrix of the transformation withrespect to these bases. Thus
α = {e1, · · · ,en} ,β = {e1, · · · ,em}
and so, in this case the coordinate maps qα and qβ are simply the identity map and therequirement that A is the matrix of the transformation amounts to
(Lb)i = ([L]b)i