Exercise 6 — Linear Algebra

Consider the two linear functions $h\colon\mathbb{R}^{3}\to\mathcal{P}_{2}$ and $g\colon\mathcal{P}_{2}\to\mathcal{M}_{2\!\times\!2}$ given as here.

\begin{pmatrix}a\\ b\\ c\end{pmatrix}\mapsto(a+b)x^{2}+(2a+2b)x+c\qquad px^{2}+qx+r\mapsto\begin{pmatrix}p&p-2q\\ q&0\end{pmatrix}

Use these bases for the spaces.

\displaystyle B=\langle\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\begin{pmatrix}0\\ 1\\ 1\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\rangle\qquad C=\langle 1+x,1-x,x^{2}\rangle

\displaystyle D=\langle\begin{pmatrix}1&0\\ 0&0\end{pmatrix},\begin{pmatrix}0&2\\ 0&0\end{pmatrix},\begin{pmatrix}0&0\\ 3&0\end{pmatrix},\begin{pmatrix}0&0\\ 0&4\end{pmatrix}\rangle

Give the formula for the composition map $g\mathbin{\circ}h\colon\mathbb{R}^{3}\to\mathcal{M}_{2\!\times\!2}$ derived directly from the above definition.
Represent $h$ and $g$ with respect to the appropriate bases.
Represent the map $g\mathbin{\circ}h$ computed in the first part with respect to the appropriate bases.
Check that the product of the two matrices from the second part is the matrix from the third part.

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