Exercise 1

Decide whether each subset of $\mathbb{R}^{3}$ is linearly dependent or linearly independent.

• (a)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ -3\\ 5\end{pmatrix},\begin{pmatrix}2\\ 2\\ 4\end{pmatrix},\begin{pmatrix}4\\ -4\\ 14\end{pmatrix}\hskip 1.00006pt\}$     (b)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 7\\ 7\end{pmatrix},\begin{pmatrix}2\\ 7\\ 7\end{pmatrix},\begin{pmatrix}3\\ 7\\ 7\end{pmatrix}\hskip 1.00006pt\}$     (c)  $\{\hskip 1.00006pt\begin{pmatrix}0\\ 0\\ -1\end{pmatrix},\begin{pmatrix}1\\ 0\\ 4\end{pmatrix}\hskip 1.00006pt\}$     (d)  $\{\hskip 1.00006pt\begin{pmatrix}9\\ 9\\ 0\end{pmatrix},\begin{pmatrix}2\\ 0\\ 1\end{pmatrix},\begin{pmatrix}3\\ 5\\ -4\end{pmatrix},\begin{pmatrix}12\\ 12\\ -1\end{pmatrix}\hskip 1.00006pt\}$