I was looking at the proof that the vector space $\wp_2 (\mathbb{R})$ can't be spanned by $2$ polynomials in the space. In their proof they picked $2$ arbitrary polynomials and equated the linear combination of the $2$ with a polynomial in the space. I've posted the proof below but can't see what they did at the end where they have the system of $3$ equations. We have a system of $3$ equations in $2$ unknowns so we should be able to solve it. Why is it unsolvable? And also, is it true that $\forall$ vector spaces of dimension $k$ that they can't be solved by $k-1$ vectors? I'm asking in the case of fnite dimensional vecctor spaces
Thanks
