Find all real solutions o f the differential equations. $\frac{d^{2} x}{d t^{2}}+2 x=\cos (t)$

If $\vec{x}$ and $\vec{y}$ are two vectors in $\mathbb{R}^{n},$ then the equation $\|\vec{x}+\vec{y}\|^{2}=\|\vec{x}\|^{2}+\|\vec{y}\|^{2}$ must hold.

Show that if A is an $n \times n$ symmetric matrix, then $(A \mathbf{x}) \cdot \mathbf{y}=\mathbf{x} \cdot(A \mathbf{y})$ for all x, y in $\mathbb{R}^{n}$.

Find the QR factorization of

$$A=\left[\begin{array}{rrr} 0 & -3 & 0 \\ 0 & 0 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & 4 \end{array}\right].$$

$A$ matrix of the form

$$A=\begin{bmatrix} a & b\\ b& -a \end{bmatrix},$$

where $a^2 + b^2 = 1,$ represents a reflection about a line. See Exercise 17. Use the formula derived in Exercise 2.1.13 to find the inverse of $A$. Explain.

In Exercise below, consider a Markov chain with state space $\{1,2, \ldots, n\}$ and the given transition matrix. Find the communication classes for each Markov chain, and state whether the Markov chain is reducible or irreducible.

$$\left[\begin{array}{ccc}1 / 4 & 1 / 2 & 1 / 3 \\ 1 / 2 & 1 / 2 & 1 / 3 \\ 1 / 4 & 0 & 1 / 3\end{array}\right]$$

Let $A$ be an $m \times m$ stochastic matrix, let $\mathbf{x}$ be in $\mathbb{R}^m$, and let $\mathbf{y}=A \mathbf{x}$. Show that

$$\left|y_1\right|+\cdots+\left|y_m\right| \leq\left|x_1\right|+\cdots+\left|x_m\right|$$

with equality holding if and only if all of the nonzero entries in $\mathbf{x}$ have the same sign.

In the Exercise below, $\mathcal{B}$ and $\mathcal{C}$ are bases for a vector space $V$. Mark the statement True or False (T/F). Justify the answer.

(T/F) If $V=\mathbb{R}^n$ and $\mathcal{C}$ is the standard basis for $V$, then$\underset{\mathcal{C} \leftarrow \mathcal{B}}{P}$ is the same as the change-of-coordinates matrix $P_{\mathcal{B}}$.

Suppose $A$ is a $4 \times 4$ matrix and $B$ is a $4 \times 2$ matrix, and let $\mathbf{u}_0 \ldots . . \mathbf{u}_3$ represent a sequence of input vectors in $\mathbb{R}^2$.

Suppose $(A, B)$ is controllable and $\mathbf{v}$ is any vector in $\mathrm{R}^4$. Explain why there exists a control sequence $\mathbf{u}_0 \ldots ., \mathbf{u}_3$ in $\mathbb{R}^2$ such that $\mathbf{x}_4=\mathbf{v}$.

If the $n \times n$ matrices A and B are orthogonal, which of the matrices must be orthogonal as well? $B^{-1}$

Mark the statement True or False. Justify the answer. $A$ represents an $n \times n$ matrix.

(T/F) If $A$ is orthogonally diagonalizable, then $A$ is symmetric.

Find, if possible, AB.

$$A=\begin{bmatrix} 1 & -1 &7\\ 2 & -1 &8 \\ 3&1&-1 \end{bmatrix}, B= \begin{bmatrix} 1& 1 &2\\ 2 & 1 &1 \\ 1&-3&2 \end{bmatrix}$$

Orthogonally diagonalize the matrices, giving an orthogonal matrix P and a diagonal matrix D.

$$\left[ \begin{array}{rr}{1} & {-5} \\ {-5} & {1}\end{array}\right]$$

The column vectors of a $5 \times 4$ matrix must be linearly dependent.