Page 81 - Advanced Linear Algebra

Linear Transformations 65

Change of Basis Matrices

Suppose that 8 and 9 ~² Á Ã Á ³ ~² Á Ã Á ³ are ordered bases for a

=
vector space . It is natural to ask how the coordinate matrices # ´ µ 8 and # ´ µ 9 are
related. Referring to Figure 2.1,
F n
I B
I (I ) -1
V C B

I C
F n

Figure 2.1

the map that takes ´#µ 8 to ´#µ 9 is 8 9 ~ Á 9 c and is called the change of basis
8
(
)
operator or change of coordinates operator . Since 89Á is an operator on

- , it has the form ( , where
(~ ² ² ³ “ Ä “ 89 Á 89 ² ³³
Á

~ ² c ²´ µ ³ “ Ä “ c ²´ µ ³³

9
9

8 8
8 8
~ ²´ µ “ Ä “ ´ µ ³³
9

9
We denote by 4 89 and call it the change of basis matrix from to .
8
(
9
,
Theorem 2.12 Let 8 and be ordered bases for a vector space

9 ~² Á Ã Á ³
= ~ . Then the change of basis operator 89 Á 9 8 c is an automorphism of - ,

whose standard matrix is
4 89, ~ ²´ µ “Ä“´ µ ³³

9
9
Hence
´#µ ~ 4 8 9 9 ´#µ 8 Á
c
and 4 ~ 98 Á 4 89 .
,
Consider the equation
(~ 4 89Á
or equivalently,
( ~ ²´ µ “Ä“´ µ ³³

9
9
(
(
(
Then given any two of an invertible d matrix Á ) 8 an ordered basis for
- (an ordered basis for - ) and 9 ), the third component is uniquely
determined by this equation. This is clear if and are given or if and are
8
9
9
(