Exercise 7

Which of these sets spans $\mathbb{R}^{3}$? That is, which of these sets has the property that any three-tall vector can be expressed as a suitable linear combination of the set’s elements?

• (a)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 2\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 3\end{pmatrix}\hskip 1.00006pt\}$     (b)  $\{\hskip 1.00006pt\begin{pmatrix}2\\ 0\\ 1\end{pmatrix},\begin{pmatrix}1\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\hskip 1.00006pt\}$     (c)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 1\\ 0\end{pmatrix},\begin{pmatrix}3\\ 0\\ 0\end{pmatrix}\hskip 1.00006pt\}$     (d)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\\ 1\end{pmatrix},\begin{pmatrix}3\\ 1\\ 0\end{pmatrix},\begin{pmatrix}-1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}2\\ 1\\ 5\end{pmatrix}\hskip 1.00006pt\}$     (e)  $\{\hskip 1.00006pt\begin{pmatrix}2\\ 1\\ 1\end{pmatrix},\begin{pmatrix}3\\ 0\\ 1\end{pmatrix},\begin{pmatrix}5\\ 1\\ 2\end{pmatrix},\begin{pmatrix}6\\ 0\\ 2\end{pmatrix}\hskip 1.00006pt\}$