c7s1 - 7.1 Coordinatization and Change of Basis 1 Chapter...

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7.1 Coordinatization and Change of Basis 1 Chapter 7. Change of Basis 7.1 Coordinatization and Change of Basis Recall. Let B = { ~ b 1 , ~ b 2 ,..., ~ b n } be an ordered basis for a vector space V . Recall that if ~v V and ~v = r 1 ~ b 1 + r 2 ~ b 2 + ··· + r n ~ b n , then the coordinate vector of ~v relative to B is ~v B =[ r 1 ,r 2 ,...,r n ]. Defnition. Let M B be the matrix having the vectors in the ordered basis B as column vectors. This is the basis matrix for B : M B = . . . . . . . . . ~ b 1 ~ b 2 ··· ~ b n . . . . . . . . . . Note. We immediately have that M B ~v B = ~v. If B 0 is another ordered basis of R n , then similarly M B 0 ~v B 0 = ~v and so ~v = M B 0 ~v B 0 = M B ~v B . Since the columns of M B 0 are basis vectors for R n (and so independent), then ~v B 0 = M 1 B 0 M B ~v B . Notice that this equation gives a relationship between the expression of ~v
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7.1 Coordinatization and Change of Basis 2 relative to basis
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c7s1 - 7.1 Coordinatization and Change of Basis 1 Chapter...

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