Exercise 8

Parametrize each subspace’s description. Then express each subspace as a span.

• The subset $\{\hskip 1.00006pt(\begin{matrix}a&b&c\end{matrix})\mid a-c=0\hskip 1.00006pt\}$ of the three-wide row vectors

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

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

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

  $\{\hskip 1.00006pt\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mid\text{$2a-c-d=0$ and $a+3b=0$}\hskip 1.00006pt\}$

• The subset $\{\hskip 1.00006pta+bx+cx^{3}\mid a-2b+c=0\hskip 1.00006pt\}$ of $\mathcal{P}_{3}$

• The subset of $\mathcal{P}_{2}$ of quadratic polynomials $p$ such that $p(7)=0$