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A question about vector spaces.

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"?

Thank you in advance
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nuodai
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Say you have two bases {v1,v2,v3}\{ v_1, v_2, v_3 \} and {w1,w2,w3}\{ w_1, w_2, w_3 \} of some three-dimensional vector space. Then because these are bases, you can write, say, w1=λ1v1+λ2v2+λ3v3w_1 = \lambda_1 v_1 + \lambda_2 v_2 + \lambda_3 v_3, 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 w1=λ1v1+λ2v2+λ3v3w_1 = \lambda_1 v_1 + \lambda_2 v_2 + \lambda_3 v_3, that's exactly the same as saying that w1=(λ1λ2λ3)w_1 = \begin{pmatrix} \lambda_1 \\ \lambda_2 \\ \lambda_3 \end{pmatrix} 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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