Exercise 4

Decide if the vector lies in the span of the set, inside of the space.

• $\begin{pmatrix}2\\ 0\\ 1\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\hskip 1.00006pt\}$, in $\mathbb{R}^{3}$

• $x-x^{3}$, $\{\hskip 1.00006ptx^{2},2x+x^{2},x+x^{3}\hskip 1.00006pt\}$, in $\mathcal{P}_{3}$

• $\begin{pmatrix}0&1\\ 4&2\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}1&0\\ 1&1\end{pmatrix},\begin{pmatrix}2&0\\ 2&3\end{pmatrix}\hskip 1.00006pt\}$, in $\mathcal{M}_{2\!\times\!2}$