Linear Algebra

Section V. Change of Basis 2371.2 Remark Perhaps a better name would be ‘change of representation matrix’but this one is standard.1.3 Lemma Left-multiplication by the change of basis matrix for B, D convertsa representation with respect to B to one with respect to D. Conversely, ifleft-multiplication by a matrix changes bases M · Rep B (⃗v) = Rep D (⃗v) then M isa change of basis matrix.Proof The first sentence holds because matrix-vector multiplication representsa map application 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 linearmap whose action is to map each vector to itself, and is therefore the identitymap.QED1.4 Example With these bases for R 2 ,( ) ( ) ( ) ( )2 1−1 1B = 〈 , 〉 D = 〈 , 〉1 01 1because( ) ( )2 −1/2Rep D ( id( )) =1 3/2the change of basis matrix is this.()−1/2 −1/2Rep B,D (id) =3/2 1/2D( ) ( )1 −1/2Rep D ( id( )) =0 1/2DFor instance, if we finding the representations of ⃗e 2( ) ( )( )0 10Rep B ( ) =Rep1 −2D ( ) =1then the matrix will do the conversion.() ( ) ( )−1/2 −1/2 1 1/2=3/2 1/2 −2 1/2()1/21/2We finish this subsection by recognizing that the change of basis matricesform a familiar set.1.5 Lemma A matrix changes bases if and only if it is nonsingular.Proof For the ‘only if’ direction, if left-multiplication by a matrix changesbases then the matrix represents an invertible function, simply because we caninvert the function by changing the bases back. Such a matrix is itself invertible,and so is nonsingular.