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Unformatted text preview: Change of Basis Matrices A basis for a vector space is not unique. In fact, given a vector space V and two different bases B = <- v 1 ,...,- v n > and C = <- w 1 ,...,- w n > we can easily construct a linear map that sends each basis vector in B to a different basis vector in C by letting- v j 7- w j and extending linearly. The purpose of this handout is to illustrate the routine for finding the matrix representation for that linear map in the case where our vector space V = R n . 1 The Easy Direction from B to E n Recall the notation for the standard basis in R n is E n = * 1 . . . , 1 . . . ,... + . Now suppose that B = <- v 1 ,...,- v n > is any other basis for R n . We may represent any vector in R n in terms of B using the notation a 1 a 2 . . . a n B := a 1- v 1 + + a n- v n . Well assume that the vectors a 1 a 2 . . . a n B and c 1 c 2 . . . c n represent the same vector in R n but written relative to the two given bases. (Think of this as these vectors have the same picture magnitude and direction.) To change a vector in terms of the basis B to a vector in terms of E n we need only do the following matrix multiplication...
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- Summer '10