Exercise 9

Find a set to span the given subspace of the given space. (Hint. Parametrize each.)

• the $xz$-plane in $\mathbb{R}^{3}$

• $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mid 3x+2y+z=0\hskip 1.00006pt\}$ in $\mathbb{R}^{3}$

• $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\mid 2x+y+w=0\text{\ and\ }y+2z=0\hskip 1.00006pt\}$ in $\mathbb{R}^{4}$

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

• The set $\mathcal{P}_{4}$ in the space $\mathcal{P}_{4}$

• $\mathcal{M}_{2\!\times\!2}$ in $\mathcal{M}_{2\!\times\!2}$