bases - Change of Basis Matrices A basis for a vector space...

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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 0 . . . , 0 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 . We’ll 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 ( -→ v 1 · · · -→ v n ) a 1 a 2 .
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