Basis and coordinates in polynomial vector space

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 $(0,1,1)$.

Determine the three basis vectors $P_1(x), P_2(x)$ and $P_3(x)$.

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 1

Solve the system $$\begin{bmatrix} \mid &\mid & \mid \\ P_1 &P_2 & P_3\\\mid&\mid&\mid\end{bmatrix} \begin{bmatrix}1&1&0\\-2&-1&1\\0&0&1\end{bmatrix} = \begin{bmatrix} 3&2&5\\2&1&0\\7&4&2 \end{bmatrix}$$

Answer 2

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 $P$?

Answer 3

You have $$\begin{pmatrix}1&-2&0 \\ 1&-1&0 \\0&1&1 \end{pmatrix}\begin{pmatrix}P_1\\P_2\\P_3\end{pmatrix}=\begin{pmatrix}Q_1\\Q_2\\Q_3\end{pmatrix}$$

Now just invert the square matrix.