Name:      

Directions: Show all work. No credit for answers without work.

1. [4 parts, 1 point each] Determine whether the following vectors are linearly independent or linearly dependent. Justify your answer.

   1. $\left[\begin{array}{c}\hfill 2\\ \hfill 3\end{array}\right],\left[\begin{array}{c}\hfill -4\\ \hfill -6\end{array}\right]$
       

   2. $\left[\begin{array}{c}\hfill 2\\ \hfill 0\\ \hfill 0\\ \hfill 0\end{array}\right],\left[\begin{array}{c}\hfill 5\\ \hfill 4\\ \hfill 0\\ \hfill 0\end{array}\right],\left[\begin{array}{c}\hfill -1\\ \hfill 5\\ \hfill -7\\ \hfill 0\end{array}\right]$
       

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   3. $\left[\begin{array}{c}\hfill 5\\ \hfill 0\\ \hfill 0\\ \hfill 0\end{array}\right],\left[\begin{array}{c}\hfill 5\\ \hfill 1\\ \hfill 0\\ \hfill 0\end{array}\right],\left[\begin{array}{c}\hfill -2\\ \hfill -3\\ \hfill 0\\ \hfill 0\end{array}\right]$
       
   4. $\left[\begin{array}{c}\hfill 1\\ \hfill 2\\ \hfill 3\end{array}\right],\left[\begin{array}{c}\hfill 4\\ \hfill 5\\ \hfill 6\end{array}\right],\left[\begin{array}{c}\hfill 7\\ \hfill 8\\ \hfill 9\end{array}\right],\left[\begin{array}{c}\hfill 10\\ \hfill 11\\ \hfill 12\end{array}\right]$
       

2. [3 points] Determine the values of $h$ that make the following vectors linearly independent.

   $$\left[\begin{array}{c}\hfill 2\\ \hfill 14\\ \hfill 2\end{array}\right],\left[\begin{array}{c}\hfill -1\\ \hfill h\\ \hfill -1\end{array}\right],\left[\begin{array}{c}\hfill 5\\ \hfill 35\\ \hfill h+1\end{array}\right]$$

3. [2 parts, 1 point each] True/False. Justify your answers.

   1. If the columns of $A$ are linearly independent, then $A𝐱=0$ has a unique solution.
   2. If $A𝐱=𝐛$ has a unique solution for at least one vector $𝐛$, then the columns of $A$ are linearly independent.
4. [1 point] Prove that if $𝐱$ and $𝐲$ are linearly independent but $\{𝐱,𝐲,𝐳\}$ is linearly dependent, then $𝐳\in \mathrm{Span}\{𝐱,𝐲\}$.