Can I get coefficients of a vector with respect to given basis for a vector space.

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Hello.

I have a question.

For example,

1. make vector space V by a basis B=[1+x,3*x,x^2] over the field of rationalsQ
2. v=3*x^2+17*x+5 in V
3. v=a*B[0]+b*B[1]+c*B[2] for some a,b,c in Q
4. [a,b,c]=?

We easily know the solution of previous example.

In general, for given basis B and a vector v, how can I get the list of coefficients?

And is it possible to the case of matrix space (B: the basis consists of matrices) ?

Thanks. :)