#### MATH 54

##### Midterm 2 | Fall '15| Nadler
1. Consider the following matrix: $A = \left[ \begin{array}{c c c} 1 & 1 & 0 \\ -1 & 3 & 0 \\ 6 & -6 & 0 \end{array} \right]$ (a) Compute the eigenvalues of $A$.

(b) Find a basis for the eigenspace corresponding to each of the eigenvalues.

(c) Is $A$ diagonalizable? Justify.

2. (a) Check that the following set of vectors is a basis for $\mathbb{R}^3$. $\mathcal{B} = \left\{ \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array}{c} 0 \\ 1 \\ 2 \end{array} \right) , \left( \begin{array}{c} 6 \\ 0 \\ 1 \end{array} \right) \right\}$

(b) Compute the change of basis matrices $P_{\mathcal{S} \leftarrow \mathcal{B}}$ and $\mathcal{P}_{\mathcal{B} \leftarrow \mathcal{S}}$ where $\mathcal{S}$ is the standard basis of $\mathbb{R}^3$.

(c) Consider the linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ given by $T \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} x - y \\ x - z \\ y - z \end{array} \right)$ Using the results of the previous part, find the matrix of $T$ with respect to $\mathcal{B}$. Feel free to write your answer as a product of matrices.

3. Label the following statement as true or false.
(a) The vector $x = \left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right]$ is an eigenvector of the matrix $A = \left[ \begin{array}{c c c} 2 & -3 & 1 \\ 4 & 0 & 3 \\ 1 & 6 & 0 \end{array} \right]$.

(b) If the $n \times n$ matrix $A$ represents the linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$ with respect to one basis, and $B$ represents the same transformation with respect to a different basis, then $\det A = \det B$.

(c) If $A$ is a $4 \times 4$ matrix with characteristic polynomial $\chi_A(\lambda) = \lambda^4 + 3\lambda^3 - 11\lambda^2 + \lambda + 5$, then $A$ must be invertible.

(d) If $W \subset \mathbb{R}^n$ is a subspace, and $y$ is in $\mathbb{R}^n$, then $y$ is in $W$ or $y$ is in $W^\bot$.

(e) For vectors $u, v \in \mathbb{R}^n$, if $||u||^2 + ||v||^2 = ||u + v||^2$, then $u$ and $v$ are orthogonal.

4. Provide an example of the following, or explain why no such example can exist. If you want to provide an example, you are allowed to choose a specfic value for $n$.
We say that a matrix $A$ is diagonalizable if $A = PDP^{-1}$ where $D$ is a diagonal matrix and $P, D$, and $P^{-1}$ are allowed to have complex entries.
(a) Two $n \times n$ matrices $A$ and $B$ where $A$ and $B$ have the same eigenvalues with the same multiplicities but are not similar.

(b) Two $n \times n$ matrices $A$ and $B$ that are diagonalizable such that $A - B$ is not diagonalizable.

(c) An $n \times n$ invertible matrix $A$ where $A$ is diagonalizable but $A^{-1}$ is not.

5. (a) Consider vectors $v_1 = \left[ \begin{array}{c} -1 \\ 2 \\ 2 \end{array} \right]$ and $v_2 = \left[ \begin{array}{c} 2 \\ -1 \\ 2 \end{array} \right]$. Find an orthonormal basis $B = \{ b_1, b_2 \}$ of the plane spanned by $v_1$ and $v_2$ in $\mathbb{R}^3$.

(b) Find a third vector $b_3$ such that the matrix $A$ with columns $b_1, b_2, b_3$ is orthogonal. (Hint: There are multiple possibilities for $b_3$.)

(c) What is $| \det(A) |$?

6. Consider the following vectors in $\mathbb{R}^4$: $v_1 = \left[ \begin{array}{c} 1 \\ -2 \\ -1 \\ 2 \end{array} \right], v_2 = \left[ \begin{array}{c} -4 \\ 1 \\ 0 \\ 3 \end{array} \right], y = \left[ \begin{array}{c} 3 \\ -1 \\ 1 \\ 13 \end{array} \right]$ Let $W = \text{Span} (v_1, v_2)$.
(a) Show that $\{ v_1, v_2 \}$ is an orthogonal set.

(b) Find the closest point to $y$ in the subspace $W$.

(c) Find the distance from $y$ to the subspace $W$.

7. An $n \times n$ square matrix $A$ is called idempotent if $A^2 = A$. Recall that $E_\lambda = \text{Nul}(A - \lambda I)$ denotes the $\lambda$-eigenspace of $A$.
(a) Show that the only possible eigenvalues of an idempotent matrix are $0$ and $1$.

(b) If $A$ is an idempotent matrix, show that $\text{Col}(A)$ is precisely the eigenspace $E_1$ of $A$.
(c) If $A$ is an idempotent matrix, show that $\text{Col}(I - A)$ is precisely the eigenspace $E_0$ of $A$. (Hint: $A - A^2 = A(I - A) = 0$.)
(d) Show that an idempotent matrix is diagonalizable. (Hint: $I = A + (I - A)$.)