Exercise 3

Find a basis for, and the dimension of, each space.

• $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\in\mathbb{R}^{4}\mid x-w+z=0\hskip 1.00006pt\}$

• the set of $5\!\times\!5$ matrices whose only nonzero entries are on the diagonal (e.g., in entry $1,1$ and $2,2$, etc.)

• $\text{$\{\hskip 1.00006pta_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\mid\text{$a_{0}+a_{1}=0$ and $a_{2}-2a_{3}=0$}\hskip 1.00006pt\}$}\subseteq\mathcal{P}_{3}$