Exercise 9

Find a basis for each space, and verify that it is a basis.

• The subspace $M=\text{$\{\hskip 1.00006pta+bx+cx^{2}+dx^{3}\mid a-2b+c-d=0\hskip 1.00006pt\}$}$ of $\mathcal{P}_{3}$.

• This subspace of $\mathcal{M}_{2\!\times\!2}$.

  $$W=\text{$\{\hskip 1.00006pt\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mid a-c=0\hskip 1.00006pt\}$}$$