Directions:

1. Write your name with one character in each box below.
2. Show all work. No credit for answers without work.
3. You are permitted to use one 8.5 inch by 11 inch sheet of prepared notes. No other aides are allowed.

1. [15 points] Determine whether the following vectors are linearly independent. If the vectors are linearly dependent, give a dependence relation.

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

2. [10 points] Characterize when the following vectors are linearly dependent in terms of simple conditions on $h$ and $k$.

   $$\left[\begin{array}{c}\hfill 2\\ \hfill 0\\ \hfill 0\\ \hfill 0\\ \hfill 0\end{array}\right],\left[\begin{array}{c}\hfill 4\\ \hfill 6\\ \hfill h\\ \hfill 0\\ \hfill 0\end{array}\right],\left[\begin{array}{c}\hfill 3\\ \hfill k\\ \hfill -h\\ \hfill 0\\ \hfill 0\end{array}\right],\left[\begin{array}{c}\hfill 1\\ \hfill 5\\ \hfill 1\\ \hfill k\\ \hfill 0\end{array}\right]$$

    

3. Transformations from ${ℝ}^2$ to ${ℝ}^2$.

   1. [9 points] Let $T_1$ be the transform given by $\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]\mapsto \left[\begin{array}{c}\hfill 0\\ \hfill x_2+1\end{array}\right]$. Is $T_1$ one-to-one/injective? Is $T_1$ onto/surjective? Is $T_1$ linear? Explain.
   2. [6 points] Let $T_2$ be the transform given by $\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]\mapsto \left[\begin{array}{c}\hfill x_1+x_2\\ \hfill x_2\end{array}\right]$ and let $T_3$ be the transformation that rotates points by $\pi ∕6$ radians. Find the standard matrix for the composition transformation $T_2\circ T_3$ given by $𝐱\mapsto T_2(T_3(𝐱))$.

4. Let $T:{ℝ}^n\to {ℝ}^m$ be a linear transform, and let $A$ be the standard matrix for $T$.

   1. [2 points] How many rows does $A$ have? How many columns?
   2. [8 points] By analyzing the pivot positions of $A$, prove that if $n>m$ then $T$ is not one-to-one/injective.

    

5. [20 points] Find the inverse of the following matrix.

   $$\left[\begin{array}{ccc}\hfill -8& \hfill 1& \hfill -9\\ \hfill 3& \hfill 0& \hfill 4\\ \hfill 1& \hfill 0& \hfill 1\end{array}\right]$$

6. [10 points] Find elementary matrices $E_1,E_2,E_3$ such that $E_3{E}_2{E}_{1}A=B$.

   $$\begin{array}{llllllll}\hfill A& =\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 7\\ \hfill 1& \hfill 2& \hfill 8\end{array}\right]& \hfill B& =\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 0\\ \hfill 2& \hfill 4& \hfill 1\end{array}\right]& \hfill & & \hfill & \end{array}$$

    

7. [10 parts, 2 points each] True/False. Assume the matrix operations below are well-defined. Justify your answers.

   1. Every elementary matrix is square.
   2. $(A+B)C=\mathit{𝐴𝐶}+\mathit{𝐵𝐶}$
   3. $(A+B)(A+B)=A^2+2\mathit{𝐴𝐵}+B^2$
   4. If $\mathit{𝐴𝐵}=\mathit{𝐵𝐴}$, then $A$ and $B$ are inverses of one another.
   5. If $A$ and $B$ are invertible $(n\times n)$-matrices, then $A$ and $B$ are row-equivalent.
   6. If $A$ and $B$ are invertible $(n\times n)$-matrices, then $A+B$ is also invertible.
   7. If $A$ and $B$ are invertible $(n\times n)$-matrices, then $\mathit{𝐴𝐵}$ is also invertible.
   8. An $(n\times n)$-matrix $A$ is invertible if and only if its transpose $A^T$ is invertible.
   9. If $S$ and $T$ are linear transforms from ${ℝ}^n$ to ${ℝ}^n$ and $S$ and $T$ are equal on at least $n$ points in ${ℝ}^n$, then $S=T$.
   10. If $T$ is a linear transform and $\{{𝐯}_1,\dots ,{𝐯}_k\}$ is linearly independent, then $\{T({𝐯}_1),\dots ,T({𝐯}_k)\}$ is also linearly independent.