Question: > 
> The vector space $P\_2(\mathbb{R})$ has a basis of $(P\_1(x),P\_2(x),P\_3(x))$ and the polynomials $Q\_1(x)=3+2x+7x^2, Q\_2(x)=2+x+4x^2, Q\_3(x)=5+2x^2$ with respect to this basis have the coordinates $(1,-2,0), \, (1,-1,0)$ and $(0,1,1)$.
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> Determine the three basis vectors $P\_1(x), P\_2(x)$ and $P\_3(x)$.
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How do I determine this? I have tried setting up a change of basis matrix $Q\_1, Q\_2, Q\_3$ as columns and then calculated the matrix vector product with each coordinate, but it wasn't correct. Not sure what to do now.

Answer: 
Solve the system
$$\begin{bmatrix} \mid &\mid & \mid \\ P\_1 &P\_2 & P\_3\\\mid&\mid&\mid\end{bmatrix} \begin{bmatrix}1&1&0\\-2&-1&1\\0&0&1\end{bmatrix} = \begin{bmatrix}
3&2&5\\2&1&0\\7&4&2
\end{bmatrix}$$
With respect to a standard basis $\{1,x,x^2\}$, you can write:

$P\left[\begin{matrix} 1 & 1 & 0 \\ -2 & -1 & 1 \\ 0 & 0 & 1 \end{matrix}\right]=\left[\begin{matrix} 3 & 2 & 5 \\ 2 & 1 & 0 \\ 7 & 4 & 2 \end{matrix}\right]$

Can you solve that for $P$?