Using isomorphisms to transform polynomials to vectors.

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Does this actually work well? We won't learn isomorphisms in linear algebra, but a friend of mine showed me an example as I prefer to work with vectors and matrices rather than polynomials (All of my problem sets are with matrices and vectors).

For example, if I wanted to find a basis for P3 that contains the polynomial 8x^3 - 2x^2 + 5x + 11, could you use isomorphisms to transform it into a vector in R4 and then find a basis?
 
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Yes, sure. This can indeed be done.

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f(ax^3+bx^2+cx+d)=(a,b,c,d)

This can be shown to be an isomorphism. So the vector spaces P_3 and \mathbb{R}^4 are the same for all linear algebra purposes. So a basis with the polynomials can be found by searching a basis in \mathbb{R}^4 first.
 

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