change of basis and composition of linear map

# change of basis and composition of linear map - V to V such...

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Suppose L is a linear map from a vector space V to another vector space W . Let ( v 1 ,...,v m ) to be a basis of V and ( w 1 ,...,w n ) to be another basis of W . Deﬁne the associated m × n matrix A to be L v 1 . . . v m = L ( v 1 ) . . . L ( v m ) = A w 1 . . . w n Let (¯ v 1 ,..., ¯ v m ) to be a new basis of V . Deﬁne the associated m × n matrix A new to be L ¯ v 1 . . . ¯ v m = L v 1 ) . . . L v m ) = A new w 1 . . . w n Then A new = BA, where B is the m × m basis change matrix from ( v 1 ,...,v m ) to (¯ v 1 ,..., ¯ v m ), i.e. B v 1 . . . v m = ¯ v 1 . . . ¯ v m We can see the A new in a new way. Let L c be the linear map from
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Unformatted text preview: V to V such that L c v 1 . . . v m = ¯ v 1 . . . ¯ v m = B v 1 . . . v m We have the following composition of linear map V L c-→ V L-→ W Let L new to be the new linear map from V to W such that it is the com-position of linear map L c and L , then the associated matrix A new of L new under the basis ( v 1 ,...,v m ) and ( w 1 ,...,w m ) is A new = BA. 1...
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## This note was uploaded on 10/03/2010 for the course MATH 223 taught by Professor Loveys during the Spring '07 term at McGill.

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