Linear Algebra, Theory And Applications, 2012a

9.3. THE MATRIX OF A LINEAR TRANSFORMATION 229
by the appropriate sized identity matrix. The requirement that A is the matrix of the
transformation amounts to
Lb = Ab
What about the situation where different pairs of bases are chosen for V and W ?How
are the two matrices with respect to these choices related? Consider the following diagram
which illustrates the situation.
F n A
−→ 2 q β2 ↓ ◦ q γ2 ↓
V −→
L W
q β1 ↑ ◦ q γ1 ↑
F n A
−→ 1 Fm
In this diagram q βi and q γi are coordinate maps as described above. From the diagram,
q −1
γ 1
q γ2 A 2 q −1
β 2
q β1 = A 1 ,
where q −1
β 2
q β1 and qγ −1
1
q γ2 are one to one, onto, and linear maps which may be accomplished
by multiplication by a square matrix. Thus there exist matrices P, Q such that P : F n → F n
and Q : F m → F m areinvertibleand
PA 2 Q = A 1 .
Example 9.3.4 Let β ≡{v 1 , ··· , v n } and γ ≡{w 1 , ··· , w n } be two bases for V . Let L
be the linear transformation which maps v i to w i . Find [L] γβ
. In case V = F n and letting
δ = {e 1 , ··· , e n } , the usual basis for F n , find [L] δ
.
∑
Letting δ ij be the symbol which equals 1 if i = j and 0 if i ≠ j, it follows that L =
i,j δ ijw i v j and so [L] γβ
= I the identity matrix. For the second part, you must have
( ) ( )
w1 ··· w n = v1 ··· v n [L]δ
and so
[L] δ
= ( ) −1 ( )
v 1 ··· v n w1 ··· w n
where ( )
w 1 ··· w n is the n × n matrix having i th column equal to w i .
Definition 9.3.5 In the special case where V = W and only one basis is used for V = W,
this becomes
q −1
β 1
q β2 A 2 q −1
β 2
q β1 = A 1 .
Letting S be the matrix of the linear transformation q −1
β 2
q β1 with respect to the standard basis
vectors in F n ,
S −1 A 2 S = A 1 . (9.3)
When this occurs, A 1 is said to be similar to A 2 and A → S −1 AS is called a similarity
transformation.
Recall the following.
Definition 9.3.6 Let S be a set. The symbol ∼ is called an equivalence relation on S if it
satisfies the following axioms.
1. x ∼ x for all x ∈ S. (Reflexive)