Compute the products $A$ $B$ and $B$ $A$, if possible, for the following:

(a) $A$

$$=\left(\begin{array}{rr}0 & -2 \\ 3 & 1\end{array}\right), B$$

$$=\left(\begin{array}{rr}-1 & 4 \\ 1 & 5\end{array}\right)$$

(b) $A$

$$=\left(\begin{array}{rrr}8 & 3 & -2 \\ 1 & 0 & 4\end{array}\right), B$$

$$=\left(\begin{array}{rr}2 & -2 \\ 4 & 3 \\ 1 & -5\end{array}\right)$$

(c) $A$

$$=\left(\begin{array}{r}0 \\ -2 \\ 4\end{array}\right), B$$

$$=\left(\begin{array}{lll}0, & -2, & 3\end{array}\right)$$

(d) $A$

$$=\left(\begin{array}{rr}-1 & 0 \\ 2 & 4\end{array}\right), \quad B$$

$$=\left(\begin{array}{rr}3 & 1 \\ -1 & 1 \\ 0 & 2\end{array}\right)$$

Verify the distributive $\mid$ law $A$ ($B$+$C$)= $AB$+$AC$ when

$$\mathbf{A}=\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rrrr} 2 & -1 & 1 & 0 \\ 3 & -1 & 2 & 1 \end{array}\right), \quad \mathbf{C}=\left(\begin{array}{rrrr} -1 & 1 & 1 & 2 \\ -2 & 2 & 0 & -1 \end{array}\right)$$

Find an equation of the tangent plane to the given parametric surface at the specified point.

$$x=u^2+1, y=v^3+1, z=u+v; (5, 2, 3)$$

Compute $A-B^T$ if

$$\begin{equation*}A=\begin{bmatrix}3&&-1&&2&&4\\1&&5&&-6&&-2\end{bmatrix},B=\begin{bmatrix}-4&&0\\ 2&&5\\ -1&&-3\\ 0&&2 \end{bmatrix}\end{equation*}$$

Write $\mathbf{u}$ as the sum of a vector parallel to $\mathbf{v}$ and a vector orthogonal to $\mathbf{v}$.

$$\mathbf{u}=3 \mathbf{j}+4 \mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}$$

A transformation T: $V_n + V_n$ is described as, T rotates every point through the same angle $\phi$ about the origin. That is, T maps a point with polar coordinates (r, 0) onto the point with polar coordinates $(r, \theta+\phi)$, where $\phi$ is fixed. Also, Tmaps 0 onto itself. Determine whether T is linear. If T is linear, describe its null space and range, and compute its nullity and rank.

Find the indicated matrices using a calculator.

$$A=\begin{bmatrix} 6 & -3 \\ 4 & -5 \end{bmatrix}$$

$$B=\begin{bmatrix} 3 & 12 \\ -9 & -6 \end{bmatrix}$$

$$C=\begin{bmatrix} -1 & 4 & -7 \\ 2 & -6 & 11 \end{bmatrix}$$

$$D=\begin{bmatrix} 7 & 9 & -6 \\ -4 & 0 & 8 \end{bmatrix}$$

$$A - B - 2C$$

Use Schwarz-Christoffel Formula to describe the image of the upper half-plane $y \geq 0$ under the conformal mapping $u^{\prime}=f(z)$ that satisfies the given conditions. Do not try to solve for $f(z)$ $f^{\prime}(z)=(z+1)^{-1 / 3}, f(-1)=0$

Graph each interval and write the interval in set-builder notation.

$$(-1, \infty)$$

Consider an indefinite quadratic form q on $\mathbb{R}^{3}$ with symmetric matrix A. If det A < 0, describe the level surface $q(\vec{x})=0.$

Write parametric equations for the given curve for the given definitions of $x$:

$$y=-4 x+1$$

a. $x=t$ b. $x=\frac{t}{2}$ c. $x=-4 t$

In its rest frame, quasar Q2203+29 produces a hydrogen emission line of wavelength 121.6 nm. Astronomers on Earth measure a wavelength of 656.8 nm for this line. Find the redshift parameter and the apparent speed of recession for this quasar.

Fill each blank with the most appropriate word from the vocabulary list below.

VOCABULARY LIST

$$\begin{array}{lll}\text { juxtaposition } & \text { puberty } & \text { primeval } \\ \text { obsolescent } & \text { abstemious } & \text { dross } \\ \text { squalid } & \text { puerile } & \text { obsolete } \\ \text { senility } & \text { immaculate } & \text { nonage } \\ \text { longevity } & \text { jetsam } & \text { carrion }\end{array}$$

In the hospital, every room was spotless. The corridors, too, were _______

The parametric curve x = 2 cos (-t), y = 2 sin (-t),0 ≤ t ≤ 2π is traced clockwise. Justify your answer.