Yes, sure. This can indeed be done.
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[tex]f(ax^3+bx^2+cx+d)=(a,b,c,d)[/tex]
This can be shown to be an isomorphism. So the vector spaces [itex]P_3[/itex] and [itex]\mathbb{R}^4[/itex] are the same for all linear algebra purposes. So a basis with the polynomials can be found by searching a basis in [itex]\mathbb{R}^4[/itex] first.
