CitiBank recorded the number of customers to use a downtown ATM during the noon hour on 32 consecutive workdays.

(a) Find the mean, median, and mode.

(b) Do these measures of center agree? Explain.

(c) Make a histogram or dot plot.

(d) Are the data symmetric or skewed? If skewed, which direction?

$$\begin{array}{rrrrrrrr}25 & 37 & 23 & 26 & 30 & 40 & 25 & 26 \\ 39 & 32 & 21 & 26 & 19 & 27 & 32 & 25 \\ 18 & 26 & 34 & 18 & 31 & 35 & 21 & 33 \\ 33 & 9 & 16 & 32 & 35 & 42 & 15 & 24\end{array}$$

Let $C=\left[c_{i j}\right]$ be an $n \times n$ positive definite symmetric matrix and let $V$ be an $n$-dimensional vector space with ordered basis $S=\left\{\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n\right\}$. For $\mathbf{v}=$ $a_1 \mathbf{u}_1+a_2 \mathbf{u}_2+\cdots+a_n \mathbf{u}_n$ and $\mathbf{w}=b_1 \mathbf{u}_1+b_2 \mathbf{u}_2+\cdots+b_n \mathbf{u}_n$ in $V$ define $(\mathbf{v}, \mathbf{w})=\sum_{i=1}^n \sum_{j=1}^n a_i c_{i j} b_j$. Prove that this defines an inner product on $V$.

Divide. Round answer to the nearest cent. $27.218$\div$ 7.6

Determine if the statement is true or false, and justify your answer. If $u_4$ is a linear combination of $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$, then $\operatorname{span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4}\right\} \neq \operatorname{span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$.

Given the standard electrode potentials.

$$\begin{align*} \begin{aligned} & \mathrm{K}^* / \mathrm{K}=-2.93 \mathrm{~V}, \mathrm{Ag}^* / \mathrm{Ag}=0.80 \mathrm{~V} \\ & \mathrm{Hg}^{2 *} / \mathrm{Hg}=0.79 \mathrm{~V} \\ & \mathrm{Mg}^{2+} / \mathrm{Mg}=-2.37 \mathrm{~V}, \mathrm{Cr}^{3 *} / \mathrm{Cr}=-0.74 \mathrm{~V} \end{aligned} \end{align*}$$

Arrange these metals in their increasing order of reducing power.

True or False: (a) A least-squares solution to an inconsistent system is a vector $\mathbf{v}$ that satisfies as many equations in the system as possibly can be satisfied. (b) For any matrix A, the matrix $\mathbf{A}^{T} \mathbf{A}$ is square and symmetric. (c) Every system $\mathbf{A x}=\mathbf{b}$ must have at least one least-squares solution. (d) If $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are both least-squares solutions to $\mathbf{A x}=\mathbf{b}$, then $\mathbf{A} \mathbf{v}_{1}=\mathbf{A} \mathbf{v}_{2}$. (e) In this section, the least-squares process is applied to solve for eigenvectors in cases in which only an estimate of the eigenvalue is known.

(a) Let P be a regular $k \times k$ stochastic matrix. Prove that the row sums of P are all equal to 1 if and only if all entries in the steady-state vector are equal to $1 / k$. (b) Use the result in part (a) to find the steady-state vector of

$$P=\left[\begin{array}{ccc}{0} & {\frac{1}{3}} & {\frac{2}{3}} \\ {\frac{1}{3}} & {\frac{2}{3}} & {0} \\ {\frac{2}{3}} & {0} & {\frac{1}{3}}\end{array}\right]$$

. (c) What can you say about the steady-state vector of a stochastic matrix that is regular and symmetric? Prove your assertion.

Under what circumstances might this trend line not be a good predictor of arm span?

In the following sentences, place a caret (^) to show where each compound word should be hyphenated. If no word In the sentence should be hyphenated, write C after the sentence.

Example: You should drink Calcium ^ enriched orange juice.

Elissa is looking for an oil free moisturizing liquid.

Este pasaje enriquece la novela de Cervantes, sobre todo porque

• presenta elementos nuevos de la vida de don Quijote y de Sancho Panza
• explica dónde encontró Cervantes la materia prima para su novela
• aclara el papel de Dulcinea del Toboso en el mundo imaginario de don Quijote
• añade a la narrativa las voces de otros autores ficticios además de la de Cervantes

Public-key cryptography is also sometimes called asymmetric cryptography. Explain why the word "asymmetric" is a sensible description of public-key cryptography, especially in comparison to symmetric-key cryptography.

Solve the problem by writing a system of equations and then solving it using the Gauss-Jordan method.

Three kinds of tea worth $\$ 4.60, \$ 5.75$, and $\$ 6.50$ per lb are to be mixed to get $20 \mathrm{lb}$ of tea worth $\$ 5.25$ per lb. The amount of $\$ 4.60$ tea used is to be equal to the total amount of the other two kinds together. How many pounds of each tea should be used?

Describe and correct the error in solving the equation.

$$\begin{aligned} 3 x - 4 & = 2 x + 1 \\ 3 x - 4 - 2 x & = 2 x + 1 - 2 x \\ x - 4 & = 1 \\ x - 4 + 4 & = 1 - 4 \\ x & = - 3 \end{aligned}$$

Let A and B be square matrices. Then A is similar to B if there is an invertible matrix M such that $B=M^{-1} A M.$ This is the relationship between matrices of a linear transformation on a vector space with respect to different bases. a. Show that similarity is reflexive: A is similar to itself. b. Show that similarity is symmetric: if A is similar to B, then B is similar to A. c. Show that similarity is transitive: if A is similar to B and B is similar to C, then A is similar to C. d. Show that similar matrices have the same eigenvalues. e. Show that similar matrices have the same determinant. f. Show that similar matrices have the same trace.