Label the following statements as true or false. In each part, V,W, and $Z$ denote vector spaces with ordered (finite) bases $\alpha, \beta,$ and $\gamma$ respectively; $T : V \rightarrow W$ and $U : W \rightarrow Z$ denote linear transformations; and $A$ and $B$ denote matrices. $\\$ (a)$[\cup T]_${\alpha}$^${\gamma}$=[T]_${\alpha}$^${\beta}$[U]_${\beta}$^${\gamma}$ $(b)$[$\mathrm{T}$(v)]_${\beta}$=[$\mathrm{T}$]_${\alpha}$^${\beta}$[v]_${\alpha} for all$v \in V $(c)$[$\mathrm{U}$(w)]_${\beta}$=[$\mathrm{U}$]_${\alpha}$^${\beta}$[w]_${\beta} for all$w \in $\mathrm{W}$ $(d)$[$\operatorname{lv}$]_${\alpha}$=I $(e)$\left[$\mathrm{T}$^{2}\right]_${\alpha}$^${\beta}$=\left([$\mathrm{T}$]_${\alpha}$^${\beta}$\right)^{2} $(e)$\left[$\mathrm{T}$^{2}\right]_${\alpha}$^${\beta}$=\left([$\mathrm{T}$]_${\alpha}$^${\beta}$\right)^{2} $(f)$A^{2}=I$implies that$A=I$or$A=-I $(g)$$\mathrm{T}$=$\mathrm{L}$_{A}$for some matrix$A .$(h)$A^{2}=O$implies that$A=O,$where$O$denotes the zero matrix.$ $(i)$L_{A+B}=L_{A}+L_{B}$.$ $(j) If$A$is square and$A_{i j}=\delta_{i j}$for all$i$and$j,$then$A=I$ .

Prove the given in $S_n$ :

Let $\alpha_1, \ldots, a_r$ be distinct even permutations, and $\beta$ an odd permutation. Then $a_1 \beta, \ldots, \alpha_r \beta$ are r distinct odd permutations.

If $\sin \alpha=-\frac{4}{5}$ and $\sec \beta=\frac{5}{3}$ for a third-quadrant angle $\alpha$ and a first-quadrant angle $\beta$, find

$$\sin (\alpha+\beta)$$

The multiple linear regression model

$$y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$$

is fitted to a data set with $n = 15$ observations, and the parameter estimates $\hat{\beta}_0 = 65.98, \hat{\beta}_1 = 23.65, \hat{\beta}_2 = 82.04$, and $\hat{\beta}_3 =17.04$ are obtained. (a) What is the fitted value of the expected value of the response variable when $x_1 = 1.5, x_2 = 1.5$, and $x_3 = 2.0$? (b) If the fitted value from part (a) has a standard error of 2.55, construct a two-sided $95\%$ confidence interval for the expected value of the response variable at this point.

Let $\alpha \in S_{n}$ for $n \geq 3$. If $\alpha \beta=\beta \alpha$ for all $\beta \in S_{n}$, prove that $\alpha$ must be the identity permutation; hence, the center of $S_{n}$ is the trivial subgroup.

Beta cutoff

Let L be the line through the origin that makes an angle $\beta$ with the positive x-axis in $R^{2}$, let $R_{\beta}$ be the standard matrix for the rotation about the origin through an angle $\beta$, and let $H_{0}$ be the standard matrix for the reflection about the x-axis. Describe the geometric effect of multiplication by $R_{\beta} H_{0} R_{\beta}^{-1}$ in terms of L.

What is the chemical symbol of oxygen?

Write a multiplication sentence with a product of -18.

Compare endorphins to neurotransmitters.

Consider a regression problem in which, for each value x of a certain variable X, the random variable $Y$ has the normal distribution with mean $\beta x$ and variance $\sigma^2$, where the values of $\beta$ and $\sigma^2$ are unknown. Show that $E(\hat{\beta})=\beta$ and $\operatorname{Var}(\hat{\beta})=\sigma^2 /\left(\sum_{i=1}^n x_i^2\right)$.

Factor: $8 x+24$ Remember to use the largest common factor and to check by multiplying. Factor out a negative factor if the first coefficient is negative.

To estimate your average monthly salary, divide your yearly salary by the number of months in a year. Write an equation to determine your yearly salary when your average monthly salary is $4560.

In this exercise, express each angle in radian measure as a multiple of $\pi$. (Do not use a calculator.)

$$-240^{\circ}$$