Linear Algebra, 2020a

Section V. Change of Basis 263
1.2 Remark A better name would be ‘change of representation matrix’ but the
above name is standard.
The next result supports the definition.
1.3 Lemma To convert from the representation of a vector ⃗v with respect to B
to its representation with respect to D use the change of basis matrix.
Rep B,D (id) Rep B (⃗v) =Rep D (⃗v)
Conversely, if left-multiplication by a matrix changes bases M · Rep B (⃗v) =
Rep D (⃗v) then M is a change of basis matrix.
Proof The first sentence holds because matrix-vector multiplication represents
a map application and so Rep B,D (id) · Rep B (⃗v) =Rep D ( id(⃗v))=Rep D (⃗v) for
each ⃗v. For the second sentence, with respect to B, D the matrix M represents
a linear map whose action is to map each vector to itself, and is therefore the
identity map.
QED
1.4 Example With these bases for R 2 ,
( ) ( ) ( ) ( )
2 1
−1 1
B = 〈 , 〉 D = 〈 , 〉
1 0
1 1
because
( ) ( )
2 −1/2
Rep D ( id( )) =
1 3/2
D
( ) ( )
1 −1/2
Rep D ( id( )) =
0 1/2
D
the change of basis matrix is this.
Rep B,D (id) =
(
)
−1/2 −1/2
3/2 1/2
For instance, this is the representation of ⃗e 2
( ) ( )
0 1
Rep B ( )=
1 −2
and the matrix does the conversion.
(
−1/2 −1/2
3/2 1/2
)(
)
1
−2
(
=
1/2
1/2
Checking that vector on the right is Rep D (⃗e 2 ) is easy.
)