Name:      

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

1. [3 points] Determine if the following vectors form a basis for ${ℝ}^4$.

   $$\left[\begin{array}{c}\hfill -3\\ \hfill 1\\ \hfill 0\\ \hfill -2\end{array}\right],\left[\begin{array}{c}\hfill 6\\ \hfill -4\\ \hfill -1\\ \hfill 6\end{array}\right],\left[\begin{array}{c}\hfill 1\\ \hfill -6\\ \hfill -3\\ \hfill 6\end{array}\right],\left[\begin{array}{c}\hfill -8\\ \hfill 1\\ \hfill 0\\ \hfill -2\end{array}\right]$$

2. [3 points] Given the matrix $A$ and an echelon form of $A$, find a basis for $\mathrm{Col}(A)$ and $\mathrm{Nul}(A)$.

   $$A=\left[\begin{array}{ccccc}\hfill 75& \hfill 383& \hfill -281& \hfill 32& \hfill 329\\ \hfill 10& \hfill 51& \hfill -37& \hfill 4& \hfill 44\\ \hfill 38& \hfill 194& \hfill -142& \hfill 16& \hfill 167\end{array}\right]\sim \left[\begin{array}{ccccc}\hfill 1& \hfill 5& \hfill -3& \hfill 0& \hfill 6\\ \hfill 0& \hfill 1& \hfill -7& \hfill 4& \hfill -1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\end{array}\right]$$

3. [2 points] Let $H$ be the set of vectors $\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\\ \hfill x_3\\ \hfill x_4\end{array}\right]\in {ℝ}^4$ such that $x_1+x_2+x_3+x_4=0$ and $x_1+2x_2+3x_3+4x_4=0$. Find a basis for $H$.

4. [2 parts, 1 point each] Subspaces.

   1. Let $H_1$ and $H_2$ be subspaces of ${ℝ}^n$. Prove that $H_1\cap H_2$ is also a subspace of ${ℝ}^n$.
   2. Give an example of subspaces $H_1$ and $H_2$ of ${ℝ}^2$ such that $H_1\cup H_2$ is not a subspace.