Let $W= \text{span}(p_1,p_2,p_3), W \subseteq R_2[x]$
$p_1(x)=1+2x+x^2$
$p_2(x)=3-9x^2$
$p_3(x)=1+4x+5x^2$
From $p_1,p_2,p_3$ choose a basis B for $W$.
I got a problem because they are asking for a basis but my solution says that they are linearly dependent... How can I get a basis anyway?
Form these polynomials to vectors, the form is $a+bx+cx^2$, so we have:
$$\vec{p_1}=\begin{pmatrix} 1\\ 2\\ 1 \end{pmatrix}, \vec{p_2}=\begin{pmatrix} 3\\ 0\\ -9 \end{pmatrix}, \vec{p_3}=\begin{pmatrix} 1\\ 4\\ 5 \end{pmatrix}$$
Now we need to check if they are linearly independent. I used determinant trick:
$$\begin{vmatrix} 1 & 3 & 1\\ 2 & 0 & 4\\ 1 & -9 & 5 \end{vmatrix}\begin{matrix} 1 & 3\\ 2 & 0\\ 1 & -9 \end{matrix}$$
If we use Saruss, indeed, the determinant is zero and thus the vectors are linearly dependent. So we cannot choose any of them as a basis..
Is there a way to get a basis anyway? Maybe I can choose one of the polynomials $p_1,p_2,p_3$ as a basis because a single one of them should be linearly independent.