8.3. THE MATRIX OF A LINEAR TRANSFORMATION 217

8.3 The Matrix of a Linear Transformation

Definition 8.3.1 In Theorem 8.2.3, the matrix of the linear transformation L ∈ L (V,W )with respect to the ordered bases β ≡ {v1, · · · , vn} for V and γ ≡ {w1, · · · , wm} for W isdefined to be [L] where [L]ij = dij . Thus this matrix is defined by L =

∑i,j [L]ij wivi. When

it is desired to feature the bases β, γ, this matrix will be denoted as [L]γβ . When there isonly one basis β, this is denoted as [L]β.

If V is an n dimensional vector space and β = {v1, · · · , vn} is a basis for V, there existsa linear map

qβ : Fn → V

defined as

qβ (a) ≡n∑

i=1

aivi

where

a =

a1...

an

 =

n∑i=1

aiei,

for ei the standard basis vectors for Fn consisting of(

0 · · · 1 · · · 0)T

. Thus the 1

is in the ith position and the other entries are 0. Conversely, if q : Fn → V is one to one,onto, and linear, it must be of the form just described. Just let vi ≡ q (ei).

It is clear that q defined in this way, is one to one, onto, and linear. For v ∈ V, q−1β (v)

is a vector in Fn called the component vector of v with respect to the basis {v1, · · · , vn}.

Proposition 8.3.2 The matrix of a linear transformation with respect to ordered bases β, γas described above is characterized by the requirement that multiplication of the componentsof v by [L]γβ gives the components of Lv.

Proof: This happens because by definition, if v =∑

i xivi, then

Lv =∑i

xiLvi ≡∑i

∑j

[L]ji xiwj =∑j

∑i

[L]ji xiwj

and so the jth component of Lv is∑

i [L]ji xi, the jth component of the matrix times the

component vector of v. Could there be some other matrix which will do this? No, because ifsuch a matrix is M, then for any x , it follows from what was just shown that [L]x =Mx.Hence [L] =M . ■

The above proposition shows that the following diagram determines the matrix of alinear transformation. Here qβ and qγ are the maps defined above with reference to theordered bases, {v1, · · · , vn} and {w1, · · · , wm} respectively.

L

β = {v1, · · · , vn} V → W {w1, · · · , wm} = γ

qβ ↑ ◦ ↑ qγFn → Fm

[L]γβ

(8.1)