Exercise 8

Find a basis for the span of each set.

• $\text{$\{\hskip 1.00006pt(\begin{matrix}1&3\end{matrix}),(\begin{matrix}-1&3\end{matrix}),(\begin{matrix}1&4\end{matrix}),(\begin{matrix}2&1\end{matrix})\hskip 1.00006pt\}$}\subseteq\mathcal{M}_{1\!\times\!2}$

• $\text{$\{\hskip 1.00006pt\begin{pmatrix}1\\ 2\\ 1\end{pmatrix},\begin{pmatrix}3\\ 1\\ -1\end{pmatrix},\begin{pmatrix}1\\ -3\\ -3\end{pmatrix}\hskip 1.00006pt\}$}\subseteq\mathbb{R}^{3}$

• $\text{$\{\hskip 1.00006pt1+x,1-x^{2},3+2x-x^{2}\hskip 1.00006pt\}$}\subseteq\mathcal{P}_{3}$

• $\text{$\{\hskip 1.00006pt\begin{pmatrix}1&0&1\\ 3&1&-1\end{pmatrix},\begin{pmatrix}1&0&3\\ 2&1&4\end{pmatrix},\begin{pmatrix}-1&0&-5\\ -1&-1&-9\end{pmatrix}\hskip 1.00006pt\}$}\subseteq\mathcal{M}_{2\!\times\!3}$