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 Mentor P: 16,518 Yes, sure. This can indeed be done. Send $$f(ax^3+bx^2+cx+d)=(a,b,c,d)$$ 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.