Linear Algebra

238 Chapter 3. Maps Between Spaces
3.V Change of Basis
Representations, whether of vectors or of maps, vary with the bases. For instance,
with respect to the two bases E2 and
� � � �
1 1
B = 〈 , 〉
1 −1
for R2 , the vector �e1 has two different representations.
� �
� �
1
1/2
RepE2 (�e1) = Rep
0
B(�e1) =
1/2
Similarly, with respect to E2, E2 and E2,B, the identity map has two different
representations.
� �
� �
1 0
1/2 1/2
RepE2,E2 (id) =
Rep
0 1
E2,B(id) =
1/2 −1/2
With our point of view that the objects of our studies are vectors and maps, in
fixing bases we are adopting a scheme of tags or names for these objects, that
are convienent for computation. We will now see how to translate among these
names—we will see exactly how representations vary as the bases vary.
3.V.1 Changing Representations of Vectors
In converting Rep B(�v) toRep D(�v) the underlying vector �v doesn’t change.
Thus, this translation is accomplished by the identity map on the space, described
so that the domain space vectors are represented with respect to B and
the codomain space vectors are represented with respect to D.
Vw.r.t. B
⏐
id�
Vw.r.t. D
(The diagram is vertical to fit with the ones in the next subsection.)
1.1 Definition The change of basis matrix for bases B,D ⊂ V is the representation
of the identity map id: V → V with respect to those bases.
⎛
.
.
⎜
.
.
RepB,D(id) = ⎜
⎝RepD(
� β1) ··· RepD( � ⎞
⎟
βn) ⎟
⎠
.
.
.
.