1. Let $A = \left[ \begin{array}{c c c} 1 & 1 & 1 \\ 0 & 1 & 0 \end{array} \right]$. Find a $3$ by $2$ matrix $B$ such that $AB = \left[ \begin{array}{c c} 1 & 0 \\ 0 & 1 \end{array} \right]$.

2. Consider the following matrix:
$A = \left[ \begin{array}{c c c c} 2 & 1 & -1 & 0 \\ 1 & 1 & 0 & 4 \\ 0 & 0 & 1 & -2 \\ 1 & 0 & 1 & 1 \end{array} \right]$
(a) Find the determinant of $A$.

(b) Is $A$ invertible? If so, find $A^{-1}$.

(c) Use parts (a) and (b) to find all solutions to $A\vec{x} = \left[ \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right]$.

3. (a) True/False: Every matrix can be row reduced into a unique row echelon form.

(b) If $A$ is a square matrix with $\det(A) = 2$, then $\det(A^{-1}) = \frac{1}{2}$.

If $T: \mathbb{R}^n \to \mathbb{R}^m$ is an injective linear transformation, then $n < m$.

(d) Any collection of $k$ vectors in $\mathbb{R}^n$ containing the zero vector is a linearly dependent set.

(e) Let $A$ be an $n$ by $n$ matrix such that $A^2 = A$, then $A$ is invertible.

4. Consider the subspace $U$ of $\mathbb{R}^5$ spanned by the set of vectors below.
$\left\{ \left( \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array} \right), \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right), \left( \begin{array}{c} 0 \\ 1 \\ 0 \\ -1 \\ 0 \end{array} \right), \left( \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array} \right), \left( \begin{array}{c} 0 \\ -5 \\ 0 \\ 5 \\ 0 \end{array} \right) \right\}$
Compute the dimension of this subspace. What is a basis for $U$?

5. (a) State the rank theorem for a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$.

(b) Compute the rank of
$A = \left( \begin{array}{c c c c} 2 & 0 & 1 & 1 \\ 3 & -1 & 1 & 2 \\ -1 & -1 & -1 & 1 \end{array} \right)$

(c) Is the linear transformation $T: \mathbb{R}^4 \to \mathbb{R}^3$ defined by $T(v) = Av$ injective? It may be helpful to use the previous parts.

(d) Use the rank theorem to show that any linear map from $\mathbb{R}^n \to \mathbb{R}^m$ cannot be injective (one-to-one) if $n > m$.

6. For each of the following provide an explicit example and show it has the desired properties:

(a) A matrix $A$ with $A^2 = 0$ but $A$ is not the zero matrix.

(a) A matrix $A$ with $A^2 = 0$ but $A$ is not the zero matrix.

(b) Matrices $A$ and $B$ such that $\det(A + B) \ne \det(A) + \det(B)$.

(c) Matrices $A$ and $B$ with $AB \ne A^TB^T$.

7. Let $\mathbb{P}_2$ be the vector space of polynomials of degree at most $2$. Let $: \mathbb{P}_2 \to \mathbb{P}_2$ be the linear transformation such that:
$T(at^2 + bt + c) = 2at + b - 2c$
That is, $T(f(t)) = \frac{df}{dt} - 2f(0)$.

(a) Write down a basis for the image of $T$.

(a) Write down a basis for the image of $T$.

(b) What is the dimension of the image of $T$?

(c) Let $H \subset \mathbb{P}_2$ be the subset of polynomials of the form $2ct + c$, where $c$ is any number. Use $T$ to show that $H$ is a subspace of $\mathbb{P}_2$.

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