Consider a linear map $f\in\mathcal L(\prod_2\Bbb R),\Bbb R^2)$ whose matrix $f_{G,B}$ in the basis $\mathcal B$ of $\prod_2(\Bbb R)$ and
$G=\{(1,-1) , (2,1)\}$ of $\Bbb R^2$ is
$$f_{G,B} = \begin{pmatrix} -10 & -2 \\ 7 & -2\\ -5 & -9 \end{pmatrix} ^T$$
Calculate $f(p)_G$ where $p$ is given by $p(x)=4x^2+8x+4 ∀x∈\Bbb R. $
Im a bit stuck on this question especially with understanding some of the notation. Because $f_{G,B}$ is in the basis B of a second degree polynomial then $B=\{1,x,x^2\}$. I know how to find the coordinated of $p$ in $B$ (its just $[ 4,8,4]^T$ but im assuming i need to find the coordinate of $p$ in G and then put into f. I might be very wrong though. Would really appreciate your help.
Cheers!
The given $3 \times 2$ matrix cannot represent a linear map in $\mathcal L(\prod_2(\Bbb R),\Bbb R^2)$ since such a matrix would have to be $2 \times 3$.
It is possible that you should transpose your matrix:
$$ f_{G,B} = \pmatrix{ -10 & 7 & -5 \\ -2 & -2 & -9}$$
In that case we have
$$f(p)_G = f_{G,B}p_B = \pmatrix{ -10 & 7 & -5 \\ -2 & -2 & -9}\pmatrix{4 \\ 8 \\ 4 } = \pmatrix{-4 \\ -60 }$$
Therefore:
$$f(p) = -4\pmatrix{1 \\ -1 } -60 \pmatrix{2 \\ 1 } = \pmatrix{-124 \\ -56}$$