Exercise 16

Find one vector $\vec{v}$ that will make each into a basis for the space.

• (a)  $\langle\begin{pmatrix}1\\ 1\end{pmatrix},\vec{v}\rangle$ in $\mathbb{R}^{2}$     (b)  $\langle\begin{pmatrix}1\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\vec{v}\rangle$ in $\mathbb{R}^{3}$     (c)  $\langle x,1+x^{2},\vec{v}\rangle$ in $\mathcal{P}_{2}$