Which of the following is the augmented matrix of the system of equations x+2y+z=-1 2x-y+3z=-4? 3x+y-z=-2 (A)

$$\begin{bmatrix}1 & 2 & 1 & -1\\2 & -1 & 3 & -4\\3 & 1 & -1 & -2\end{bmatrix}$$

(B)

$$\begin{bmatrix}1 & 2 & 1 & 1\\2 & -1 & 3 & 4\\3 & 1 & -1 & 2\end{bmatrix}$$

(C)

$$\begin{bmatrix}1 & 2 & 1 & 0\\2 & -1 & 3 & 0\\3 & 1 & -1 & 0\end{bmatrix}$$

(D)

$$\begin{bmatrix}1 & 2 & 1\\2 & -1 & 3\\3 & 1 & -1\end{bmatrix}$$

(E)

$$\begin{bmatrix}1 & 2 & -1\\2 & -1 & -3\\3 & 1 & 1\end{bmatrix}$$

Two vertices of a figure are $(-4,0)$ and $(2,0)$.

a. Name the coordinates of the third vertex above the $x$-axis if the figure is an equilateral triangle. (Hint: One of the coordinates is irrational.)

Define the linear transformation $T: R^{n} \rightarrow R^{m}$ by T(v) = Av. Use the matrix A to (a) determine the dimensions of $R^n$ and $R^m$, (b) find the image of v, and (c) find the preimage of w.

$$A=\left[\begin{array}{ll} {2} & {1} \\ {0} & {1} \end{array}\right], \mathbf{v}=(8,4), \mathbf{w}=(5,2)$$

Find A + B.

$$A=\begin{bmatrix} 0 & 1&2\\ 1&-1& 0\\ 2&1& 1 \end{bmatrix}, \quad B=\begin{bmatrix} 0 & 1&-1\\ 2&1& 1\\ 0&1& 1 \end{bmatrix}$$

Underline the verb in each sentence. Then, above the verb, write AV if the verb is an action verb or LV if it is a linking verb.

Example 1. The tree $\overset{\textit{\color{#c34632}{LV}}}{{\underline{\text{grew}}}}$ tall and sturdy.

I smell the aroma of a freshly mowed lawn.

An ordered pair that makes an equation in two variables true is called a $\underline {?}$ .

Solve the boundary-value problem, if possible.

$$y’’+4y=0, y(0)=5, y(π/4)=3$$

Simplify. Give any restrictions on the variables. $\frac{5 a-10}{a-2}$

Simplify. The variables represent any real number.

$$\sqrt{a^2}$$

Let A denote an

$$m \times n$$

matrix. (a) Show that null A=null(UA) for every invertible

$$m \times n$$

matrix U. (b) Show that dim(null A)=dim(null(AV)) for every invertible

$$n \times n$$

matrix V. [Hint: If

$$\left\{ \mathbf { x } _ { 1 } , \mathbf { X } _ { 2 } , \dots , \mathbf { X } _ { k } \right\}$$

is a basis of null A, show that

$$\left\{ V ^ { - 1 } \mathbf { x } _ { 1 } , V ^ { - 1 } \mathbf { x } _ { 2 } , \dots , V ^ { - 1 } \mathbf { x } _ { k } \right\}$$

is a basis of null(AV).]

A system is shown in Figure (a). The response to a unit step, when $K=1$, is shown in Figure b). Determine the value of $K$ so that the steadystate error is equal to zero.

Define the linear transformation T by T(v) = Av. Find (a) ker(T) , (b) nullity(T) , (c) range(T) , and (d) rank(T).

$$A=\left[\begin{array}{rr} {1} & {2} \\ {-1} & {0} \\ {1} & {1} \end{array}\right]$$

Find the lexicographic ordering of these n-tuples: a) (1, 1, 2), (1, 2, 1) b) (0, 1, 2, 3), (0, 1, 3, 2) c) (1, 0, 1, 0, 1), (0, 1, 1, 1, 0)

Find the rational number, expressed in lowest terms, represented by each of the following simple continued fractions. a) [2; 7], b) [1; 2, 3], c) [0; 5 , 6], d) [3; 7, 15, 1], e) [1; 1], f) [1; 1, 1], g) [1; 1, 1, 1], h) [1; 1, 1, 1, 1].