Linear Algebra

236 Chapter Three. Maps Between SpacesVChange of BasisRepresentations vary with the bases. For instance, ⃗e 1 ∈ R 2 has two differentrepresentations( )( )11/2Rep E2(⃗e 1 ) = Rep0B (⃗e 1 ) =1/2with respect to the standard basis and this one.( ) ( )1 1B = 〈 , 〉1 −1The same is true for maps; with respect to the basis pairs E 2 , E 2 and E 2 , B, theidentity map has two different representations.( )( )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, byfixing bases we are adopting a scheme of tags or names for these objects thatare convenient for calculations. We will now see how to translate among thesenames, so we will see exactly how the 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 and thecodomain space vectors are represented with respect to D.V wrt Bid⏐↓V wrt 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 ) ⎟⎠..