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Q1. Find the three projectors in the first picture in this lecture note.
$\mathbf{P}_1$ projects into $\text{span}\left(\left\{\begin{pmatrix} 1 \\ 2\end{pmatrix}\right\}\right)$ along $\text{span}\left(\left\{\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right\}\right)$.
$\mathbf{P}_2$ projects into $\text{span}\left(\left\{\begin{pmatrix} 1 \\ 2\end{pmatrix}\right\}\right)$ along $\text{span}\left(\left\{\begin{pmatrix} 1 \\ 0 \end{pmatrix}\right\}\right)$.
$\mathbf{P}_3$ is the orthogonal projection into $\text{span}\left(\left\{\begin{pmatrix} 1 \\ 2\end{pmatrix}\right\}\right)$.
Q2. Let $\mathbf{A} \in \mathbb{R}^{n \times n}$ be a symmetric matrix. Prove that $\langle \mathbf{A} \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{x}, \mathbf{A} \mathbf{y} \rangle$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$. Give an example that it is not necessary true if $\mathbf{A}$ is not symmetric.
Q3. Find the orthogonal projection of the point $\mathbf{1}_3$ into the plane spanned by the vectors $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} -2 \\ 2 \\ 1 \end{pmatrix}$.
Q4. Matrices that satisfy $\mathbf{A}' \mathbf{A} = \mathbf{A} \mathbf{A}'$ are said to be normal. Give an example of asymmetric (not symmetric), normal matrix. If $\mathbf{A}$ is normal, then prove that every vector in $\mathcal{C}(\mathbf{A})$ is orthogonal to every vector in $\mathcal{N}(\mathbf{A})$.
Q5. For $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$, show that $(\mathbf{x} + \mathbf{y}) \perp (\mathbf{x} - \mathbf{y})$ if and only if $\|\mathbf{x}\| = \|\mathbf{y}\|$.
Q6. Let $\mathbf{A}$ be a symmetric matrix. Show that the system $\mathbf{A} \mathbf{x} = \mathbf{b}$ has a solution if and only if $\mathbf{b}$ is orthogonal to $\mathcal{N}(\mathbf{A})$.
Q7. Find an orthonormal basis for each of the four fundamental subspaces of
$$
\mathbf{A} = \begin{pmatrix}
2 & -2 & -5 & -3 & -1 & 2 \\
2 & -1 & -3 & 2 & 3 & 2 \\
4 & -1 & -4 & 10 & 11 & 4 \\
0 & 1 & 2 & 5 & 4 & 0
\end{pmatrix}.
$$
Hint: Gram-Schmidt algorithm can be helpful.
Q8. Let $\mathbf{A}, \mathbf{B} \in \mathbb{R}^{n \times n}$ be two orthogonal matrices. Show that $\mathbf{A} \mathbf{B}$ is an orthogonal matrix. Construct an example to show that $\mathbf{A} + \mathbf{B}$ need not be orthogonal.
Q9. Determinant.
(1). Find the determinant of the following two matrices without doing any computations:
$$
\begin{pmatrix}
0 & 0 & 1\\
0 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix}, \quad \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.
$$
(2). Let $\mathbf{A} \in \mathbb{R}^{5 \times 5}$ with $\det(\mathbf{A}) = -3$. Find $\det(\mathbf{A}^3)$, $\det(\mathbf{A}^{-1})$, and $\det(2\mathbf{A})$.
(3). Find the determinant of the matrix
$\begin{pmatrix}
0 & 0 & 1 \\
2 & 3 & 4 \\
0 & 5 & 6
\end{pmatrix}$. Hint: find the row and column permutations that make $\mathbf{A}$ triangular; then use product rule.
(1). Find eigenvalues and eigenvectors of $\mathbf{A}$ and $\mathbf{A}^{-1}$. Do they have same eigenvectors? What's the relationship between their eigenvalues?
(2). Find eigenvalues of $\mathbf{B}$ and $\mathbf{A} + \mathbf{B}$. Are eigenvalues of $\mathbf{A} + \mathbf{B}$ equal to eigenvalues of $\mathbf{A}$ plus eigenvalues of $\mathbf{B}$?
(3). Find eigenvalues of $\mathbf{A} \mathbf{B}$ and $\mathbf{B} \mathbf{A}$. Are the eigenvalues of $\mathbf{A} \mathbf{B}$ equal to eigenvalues of $\mathbf{A}$ times eigenvalues of $\mathbf{B}$? Are the eigenvalues of $\mathbf{A} \mathbf{B}$ equal to eigenvalues of $\mathbf{B} \mathbf{A}$?