Linear Algebra

232 Chapter Three. Maps Between SpacesVChange of BasisRepresentations, whether of vectors or of maps, vary with the bases. For instance,with respect to the two bases E 2 and( (1 1B = 〈 , 〉1)−1)for R 2 , the vector ⃗e 1 has two different representations.( ( )1 1/2Rep E2(⃗e 1 ) = Rep0)B (⃗e 1 ) =1/2Similarly, with respect to E 2 , E 2 and E 2 , B, the identity map has two differentrepresentations.( )( )1 01/2 1/2Rep E2 ,E 2(id) =Rep0 1E2 ,B(id) =1/2 −1/2With our point of view that the objects of our studies are vectors and maps, infixing bases we are adopting a scheme of tags or names for these objects, thatare convienent for computation. We will now see how to translate among thesenames — we will see exactly how representations vary as the bases vary.V.1 Changing Representations of VectorsIn converting Rep B (⃗v) to Rep D (⃗v) the underlying vector ⃗v doesn’t change.Thus, this translation is accomplished by the identity map on the space, describedso that the domain space vectors are represented with respect to B andthe codomain space vectors are represented with respect to D.V w.r.t. B⏐id↓V w.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 representationof the identity map id: V → V with respect to those bases.⎛⎞..Rep B,D (id) = ⎜⎝Rep D ( β ⃗ 1 ) · · · Rep D ( β ⃗ n ) ⎟⎠..