446 CHAPTER 18. LINEAR TRANSFORMATIONS

Now suppose 5.) and suppose Lv = Lw. Then L(v−w) = 0 and so by 5.), v−w = 0showing that L is one to one. ■

Also it is important to note that composition of linear transformation corresponds to-multiplication of the matrices. Consider the following diagram.

X A−→

Y B−→

Z

qα ↑ ◦ ↑ qβ ◦ ↑ qγ

Fn [A]βα

−→Fm [B]

γβ

−→Fp

where A and B are two linear transformations, A∈L (X ,Y ) and B∈L (Y,Z) . Then B◦A∈L (X ,Z) and so it has a matrix with respect to bases given on X and Z, the coordinate mapsfor these bases being qα and qβ respectively. Then

B◦A = qγ [B]γβq−1

βqβ [A]βα

q−1α = qγ [B]γβ

[A]βα

q−1α .

But this shows that [B]γβ

[A]βα

plays the role of [B◦A]γα

, the matrix of B ◦ A. Hencethe matrix of B ◦A equals the product [B]

γβ[A]

βα. Of course it is interesting to note that

although [B◦A]γα

must be unique, the matrices, [B]γβ

and [A]βα

are not unique becausethey depend on the basis chosen for Y .

Theorem 18.5.17 The matrix of the composition of linear transformations equals the prod-uct of the matrices of these linear transformations in the same order as the composition.

18.5.1 Some Geometrically Defined Linear TransformationsThis is a review of earlier material. If T is any linear transformation which maps Fn to Fm,there is always an m×n matrix A with the property that

Ax = T x (18.12)

for all x ∈ Fn. How does this relate to what is discussed above? In terms of the abovediagram,

{e1, · · · ,en} Fn T−→

Fm {e1, · · · ,en}

qFn ↑ ◦ ↑ qFm

Fn M−→

Fm

where

qFn (x)≡n

∑i=1

xiei = x.

Thus those two maps are really just the identity map. Thus, to find the matrix of the lineartransformation T with respect to the standard basis vectors,

T ek = Mek

In other words, the kth column of M equals T ek as noted earlier. All the earlier considera-tions apply. These considerations were just a specialization to the case of the standard basisvectors of this more general notion which was just presented.