8.3. THE MATRIX OF A LINEAR TRANSFORMATION 219

which illustrates the situation.Fn A2−→ Fm

qβ2↓ ◦ qγ2

↓V L−→ W

qβ1↑ ◦ qγ1

↑Fn A1−→ Fm

In this diagram qβiand qγi

are coordinate maps as described above. From the diagram,

q−1γ1qγ2

A2q−1β2qβ1

= A1,

where q−1β2qβ1

and q−1γ1qγ2

are one to one, onto, and linear maps which may be accomplishedby multiplication by a square matrix. Thus there exist matrices P,Q such that P : Fn → Fn

and Q : Fm → Fm are invertible and

PA2Q = A1.

Example 8.3.4 Let β ≡ {v1, · · · ,vn} and γ ≡ {w1, · · · ,wn} be two bases for V . Let Lbe the linear transformation which maps vi to wi. Find [L]γβ . In case V = Fn and lettingδ = {e1, · · · , en} , the usual basis for Fn, find [L]δ.

Letting δij be the symbol which equals 1 if i = j and 0 if i ̸= j, it follows that L =∑i,j δijwivj and so [L]γβ = I the identity matrix. For the second part, you must have(

w1 · · · wn

)=(

v1 · · · vn

)[L]δ

and so

[L]δ =(

v1 · · · vn

)−1 (w1 · · · wn

)where

(w1 · · · wn

)is the n× n matrix having ith column equal to wi.

Definition 8.3.5 In the special case where V = W and only one basis is used for V = W,this becomes

q−1β1qβ2

A2q−1β2qβ1

= A1.

Letting S be the matrix of the linear transformation q−1β2qβ1

with respect to the standard basisvectors in Fn,

S−1A2S = A1. (8.3)

When this occurs, A1 is said to be similar to A2 and A → S−1AS is called a similaritytransformation.

Recall the following.

Definition 8.3.6 Let S be a set. The symbol ∼ is called an equivalence relation on S if itsatisfies the following axioms.

1. x ∼ x for all x ∈ S. (Reflexive)

2. If x ∼ y then y ∼ x. (Symmetric)

3. If x ∼ y and y ∼ z, then x ∼ z. (Transitive)