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# A question about vector spaces. watch

1. In P_sub 2, find the change-of-coordinates matrix from the basis B= {1-3t^2, 2+t-5t^2, 1+2t} to the standard basis. Then write t^2 as a linear combination of the polynomials in B.

I know how to find the change-of-coordinates matrix from the basis B to the standard matrix...but I'm a bit confused about writing t^2 as a linear combination of the polynomials in B. My question: Does "write t^2 as a linear combination of the polynomials in B" mean the same thing as "find the B-coordinate vector for t^2"?

2. Say you have two bases and of some three-dimensional vector space. Then because these are bases, you can write, say, , as a linear combination of vectors in the 'v' basis. You're doing exactly the same thing here. The reason why it's the same as finding the B-coordinate vector is because when you write , that's exactly the same as saying that with respect to the other basis. The same thing carries over to polynomials (because polynomials and vectors are basically the same thing).

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Updated: April 4, 2011
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