5.2. THE MATRIX OF A LINEAR TRANSFORMATION 107

What about the situation where different pairs of bases are chosen for V and W? Howare the two matrices with respect to these choices related? Consider the following diagramwhich illustrates the situation.

Fn A2−→ Fm

qβ 2↓ ◦ qγ2 ↓V L−→ W

qβ 1↑ ◦ qγ1 ↑Fn A1−→ Fm

In this diagram qβ iand qγ i are coordinate maps as described above. From the diagram,

q−1γ1

qγ2A2q−1β 2

qβ 1= A1,

where q−1β 2

qβ 1and q−1

γ1qγ2 are one to one, onto, and linear maps which may be accom-

plished by multiplication by a square matrix. Thus there exist matrices P,Q such thatP : Fn→ Fn and Q : Fm→ Fm are invertible and

PA2Q = A1.

Example 5.2.3 Let β ≡ {v1, · · · ,vn} and γ ≡ {w1, · · · ,wn} be two bases for V . Let L bethe linear transformation which maps vi to wi. Find [L]

γβ.

Letting δ i j be the symbol which equals 1 if i = j and 0 if i ̸= j, it follows that L =

∑i, j δ i jwiv j and so [L]γβ

= I the identity matrix.

Definition 5.2.4 In the special case where V = W and only one basis is used for V = W,this becomes

q−1β 1

qβ 2A2q−1

β 2qβ 1

= A1.

Letting S be the matrix of the linear transformation q−1β 2

qβ 1with respect to the standard

basis vectors in Fn,S−1A2S = A1. (5.5)

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

Recall the following.

Definition 5.2.5 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)

Definition 5.2.6 [x] denotes the set of all elements of S which are equivalent to x and [x] iscalled the equivalence class determined by x or just the equivalence class of x.