Let $P\_3$ be the set of all real polynomials of degree 3 or less. This set forms a real vector space. Show that $\{2x^3+x+1,x−2,x^3−x^2\}$ is a linearly independent set, and ﬁnd a basis for $P\_3$ which includes these three polynomials.

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Linear independence is easy, I just put the coefficients in a set and rref;

$\begin{bmatrix}2&1&1\\0&1&-2\\1&-1&0\end{bmatrix} => rref =>\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

So I've proven that the set is linearly independent. Also I have 3 pivots for 3 variables, so think I can write the vector space $P\_3$ as a span of the polynomials (or, vectors of the coefficients);

$P\_3 = span\{(2, 1, 1),(0, 1,-2),(1, -1, 0)\}$

Since a basis is simply a set of linearly independent vectors which cover the span, are these three vectors a basis for $P\_3$, or do I have more work to do? (EDIT: That is, if we pretend that I hadn't completely erased an entire degree from the polynomials).

Also, given that $P\_3$ is talked about in terms of polynomials (and assuming I've done the above properly); should I answer questions like this using the polynomials themselves, or is it okay to simply use vectors/matrices/etc with the coefficients?