Let V be the linear space of polynomials of degree ≤ 2 and let T : V → W be given by T(p) = q where q is a polynomial such that q(x) = Z x 0 p(s)ds. Is T linear? If so, for what W? If T is linear then derive its matrix of the linear map.

Known Matrix:"A=\\begin{bmatrix}\n 2 & 1 & 2 \\\\\n 1 & 2 & 2 \\\\\n 1 & 1 & 3\n\\end{bmatrix}""B=\\begin{bmatrix}\n 3 & 0 & 2 \\\\\n 0 & 1 & a \\\\\n 0 & 2 & 2a\n\\end{bmatrix}"Determine the characteristic equation det(A − λI) = 0 of the matrix aboveDetermine the eigenvalues of the matrix and the basis of the eigenspaceNote: In matrix B, let the value of a be so that the eigenvalues and the basis of the eigenspace are dependent on a.

$\begin{array}{l}A=\left[\begin{array}{lll}a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a\end{array}\right], \text { where } a \in \mathbb{F} \text {. Prove that } \\ A^{n}=\left[\begin{array}{ccc}a^{n} & n a^{n-1} & \frac{n(n-1)}{2} a^{n-2} \\ 0 & a^{n} & n a^{n-1} \\ 0 & 0 & a^{n}\end{array}\right] \quad \text { for all integers } n \geq 1 .\end{array}$

Find the plane tangent to xyz = α at the point (β, β, β).

Let f(x)=7x+5 and g(x)=4x−7. Find (f+g)(x), (f−g)(x), (fg)(x), and fg(x). Give the domain of each.

1. Let $y=f(x)$ be given by the graph below. Sketch the graph of each of the following. Label at least 3 points on each graph. a. $y=f(2 x)+1$ b. $y=-f(x+2)$

If $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ are vectors in three-dimensional space, which of the following are true? $(3 \mathbf{u}) \times \mathbf{v}=3(\mathbf{u} \times \mathbf{v})$ $(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}=\mathbf{0}$ $\mathbf{w} \times \mathbf{w}=\mathbf{0}$ $(\mathbf{u}+\mathbf{v}) \times \mathbf{w}=\mathbf{u} \times \mathbf{w}+\mathbf{v} \times \mathbf{w}$ $\mathbf{u} \times \mathbf{v}=\mathbf{v} \times \mathbf{u}$ $\mathbf{u} \times(\mathbf{v}+\mathbf{w})=\mathbf{u} \times \mathbf{v}+\mathbf{u} \times \mathbf{w}$ $\mathbf{u} \times \mathbf{v}=-\mathbf{v} \times \mathbf{u}$ $(\mathbf{u} \times \mathbf{v}) \times \mathbf{u}=\mathbf{0}$ $\boldsymbol{v} \mathbf{u} \times \mathbf{0}=\mathbf{0}$.

can someone please help You are given the cross product \[ \left(\begin{array}{l} 3 \\ 1 \\ 4 \end{array}\right) \times\left(\begin{array}{l} 4 \\ 0 \\ 2 \end{array}\right)=\left(\begin{array}{c} 2 \\ 10 \\ -4 \end{array}\right) \] Let's observe that \[ \begin{aligned} \left(\begin{array}{l} 3 \\ 1 \\ 4 \end{array}\right) \cdot\left(\begin{array}{c} 2 \\ 10 \\ -4 \end{array}\right)=\text { Number } \\ \text { and }\left(\begin{array}{l} 4 \\ 0 \\ 2 \end{array}\right) \cdot\left(\begin{array}{c} 2 \\ 10 \\ -4 \end{array}\right)=\text { Number } \end{aligned} \] This means that $\left(\begin{array}{c}2 \\ 10 \\ -4\end{array}\right)$ is perpendicular $\quad$ to both $\left(\begin{array}{l}3 \\ 1 \\ 4\end{array}\right)$ and $\left(\begin{array}{l}4 \\ 0 \\ 2\end{array}\right)$, the vectors in the given cross product. i) So, using this idea, a (non-zero) vector perpendicular to both $\left(\begin{array}{l}0 \\ 3 \\ 5\end{array}\right)$ and $\left(\begin{array}{l}-2 \\ -5 \\ -1\end{array}\right)$ is ii) Likewise, a (non-zero) vector $\mathbf{v}$ satisfying both $\mathbf{v} \cdot\left(\begin{array}{l}1 \\ 3 \\ 1\end{array}\right)=0$ and $\mathbf{v} \cdot\left(\begin{array}{c}1 \\ 2 \\ -2\end{array}\right)=0$ is 因国。

Let ? ∶ ℝ3 →ℝ3 be the transformation that reflects each vector x = (?1,?2,?3) through the plane ?3 = 0 onto T (x) = (?1,?2,−?3). Show that T is a linear transformation. Please solve and explain each step please. Show work aswell

Elementary Matrices: Problem 1 Previous Problem Problem List Next Problem (1 point) a. Suppose that: \[ E_{1}\left[\begin{array}{cc} -4 & 3 \\ -4 & -1 \end{array}\right]=\left[\begin{array}{ll} -16 & 12 \\ -4 & -1 \end{array}\right] \] Find $E_{1}$ and $E_{1}^{-1}$ b. Suppose that: \[ E_{2}\left[\begin{array}{cc} -4 & 3 \\ -4 & -1 \end{array}\right]=\left[\begin{array}{ll} -4 & -1 \\ -4 & 3 \end{array}\right] \] Find $E_{2}$ and $E_{2}^{-1}$ \[ E_{2}=[ \] c. Suppose that: \[ E_{3}\left[\begin{array}{cc} -4 & 3 \\ -4 & -1 \end{array}\right]=\left[\begin{array}{ll} -16 & 0 \\ -4 & -1 \end{array}\right] \] Find $E_{3}$ and $E_{3}^{-1}$ d. Suppose that: f. Suppose that: \[ E_{6}\left[\begin{array}{ccc} -3 & 4 & 5 \\ 4 & 5 & -2 \\ 5 & 1 & 1 \end{array}\right]=\left[\begin{array}{ccc} -3 & 4 & 5 \\ 4 & 5 & -2 \\ -16 & 29 & 36 \end{array}\right] \] Find $E_{6}$ and $E_{6}^{-1}$

Two highways meet at $53^{\circ}$. If two bike riders take separate routes at $17 \mathrm{~km}$ per hour and $24 \mathrm{~km}$ per hour how far apart are they after two hours?

A 24-centimeter tall vase has cross sections parallel to the base that are circles. Each circular cross section $h$ centimeters above the base has a radius of $r(h)=8-0.1 h+2 \sin \left(0.3 h^{0.9}\right)$ centimeters for $0 \leq h \leq 24$. A sketch of the vase and the graph of $r$ are shown above. (a) Find the area of the region between the graph of $r$ and the $h$-axis from $h=0$ to $h=24$. (b) Find the volume of the vase. (c) Evaluate $\int_{4}^{20} r^{\prime}(h) d h$. Using correct units, interpret the meaning of $\int_{4}^{20} r^{\prime}(h) d h$ in the context of the problem. (d) Water is poured into the vase. At the instant when the depth of the water in the vase is 10 centimeters, the depth of the water is increasing at a rate of 0.6 centimeter per second. At that instant, what is the rate of change of the circumference of the surface of the water, in centimeters per second?convince me pls

Question The graph of the polynomial $f(x)$ is given below. If $f(x)$ has degree 4 , find the factored equation for $f(x)$. Provide your answer below: \[ f(x)= \] D FEEDBACK MORE INSTRUCTION SUBMIT Content attribution

A fence is to be built to enclose a rectangular area of 320 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 14 dollars per foot. Find the dimensions of the enclosure that is most economical to construct. Dimensions: --------- X ----------

The campus of a college has plans to construct a rectangular parking lot on land bordered on one side by a highway. There are $640 \mathrm{ft}$ of fencing available to fence the other three sides. Let $x$ represent the length of each of the two parallel sides of fencing. (a) Express the length of the remaining side to be fenced in terms of $x$. (b) What are the restrictions on $x$ ? (c) Determine a function A that represents the area of the parking lot in terms of $x$. (d) Determine the values of $x$ that will give an area between 30,000 and 40,000 $\mathrm{ft}^{2}$. (e) What dimensions will give a maximum area, and what will this area be? (a) The length of the remaining side to be fenced in terms of $x$ is (Do not factor.)

(a) Express $\sin x+2 \cos x$ in the form $R \sin (x+\alpha)$ where $R$ and $\alpha$ are constants, $R>0$ and $0<\alpha<\frac{\pi}{2}$ Give the exact value of $R$ and give the value of $\alpha$ in radians to 3 decimal places. (3) The temperature, $\theta^{\circ} \mathrm{C}$, inside a room on a given day is modelled by the equation \[ \theta=5+\sin \left(\frac{\pi t}{12}-3\right)+2 \cos \left(\frac{\pi t}{12}-3\right) \quad 0 \leqslant t<24 \] where $t$ is the number of hours after midnight. Using the equation of the model and your answer to part (a), (b) deduce the maximum temperature of the room during this day, (c) find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

A company’s sales are seasonal with the peak in mid-December and the lowest point in mid-June. The company makes $100,000 in sales in December and only $20,000 in June. (a) Find a trigonometric function, s = f(t), representing sales at time t months after mid-January. (b) What would you expect the sales to be for mid-April? (c) Find the t-values for which s = 60,000

Consider the following LP problem: Maximize earnings = $0.80X1 + $0.40X2 + $1.20X3 - $0.10X4 subject to X1 + 2X2 + X3 + 5X4 ≤ 150 X2 - 4X3 + 8X4 = 70 6X1 + 7X2 + 2X3 - X4 ≥ 120 X1, X2, X3, X4 ≥ 0 Q: The constraint X2 - 4X3 + 8X4 = 70, when converted to an = constraint for use in the simplex algorithm, will be

CLO 5 - You must answer these three sub-questions 5.1 Safety and Quality in healthcare should be “driven by information”. Discuss this statement in the context of Antibiotic Stewardship and increasing patient safety in the healthcare environment. (250 words, 20 marks) 5.2 The Victorian Government has recently mandated COVID vaccination for anyone working in healthcare services. Discuss this mandate from two different ethical perspectives. (125 words, 10 marks) 5.3 Anna is a 76-year-old widow who is receiving care in your rehabilitation ward after suffering a stroke that has left her paralysed down her right side. With some speech pathology support, Anna is able to answer simple “yes” and “no” questions and can write some words with her left hand. Her family believes that Anna must be protected from any emotional or psychological distress. According to Anna’s family, this belief is part of their cultural and religious background. Her family is expressing their love and respect by shielding her from the true news about her poor prognosis. At the family’s insistence, Anna’s doctor requests a physiotherapy assessment to determine her ability to participate in rehabilitation therapy, with the goal of helping her to walk again. During her assessment of Anna, the physiotherapist realizes that Anna has little or no understanding of the true state of her health. Anna says that she is only resting after the stroke and says that she is looking forward to going home, sleeping in her own bed and working in the garden. The physiotherapist does not believe that this will be possible, and Anna will probably need to be admitted to an Aged Care Facility or live with one of her children in a modified house. The physiotherapist feels very distressed that this patient has little or no understanding of the gravity of her condition and the implications this has for her future health challenges and appropriate care options. The physiotherapist approaches you as the Nursing Unit Manager. You realise that you have an ethical issue to think about. State the decision you would make in this case and justify your decision using two of the following five approaches to ethical decision making. (125 words, 10 marks)

Identify the domain and range of the function shown in the graph below using interval notation. Select the correct answer below: Domain $=[-2,2] ;$ Range $=[-2,2]$ Domain $=[-2,2) ;$ Range $=(-\infty, 2)$ Domain $=(-2,2) ;$ Range $=[-2,2)$ Domain $=[-2,2) ;$ Range $=[-2,2)$

*10.60 Consider the CMOS inverter circuit shown in Figure 10.59a. Ideal n- and p-channel devices are to be designed with channel lengths of $L=2.5 \mu \mathrm{m}$ and oxide thicknesses of $t_{\mathrm{ox}}=450 \AA$. Assume the inversion channel mobilities are one-half the bulk values. The threshold voltages of the $\mathrm{n}$ - and $\mathrm{p}$-channel transistors are to be $+0.5 \mathrm{~V}$ and $-0.5 \mathrm{~V}$, respectively. The drain current is to be $I_{D}=0.256 \mathrm{~mA}$ when the input voltage to the inverter is $1.5 \mathrm{~V}$ and $3.5 \mathrm{~V}$ with $V_{D D}=5 \mathrm{~V}$. The gate material is to be the same in each device. Determine the type of gate, substrate doping concentrations. and channel widths. 1. For two $n \times n$ matrices $\mathbf{A}$ and $\mathbf{B}$, prove that (a) $(\mathbf{A B})^{\top}=\mathbf{B}^{\top} \mathbf{A}^{\top}$; (b) $(\mathbf{A B})^{\dagger}=\mathbf{B}^{\dagger} \mathbf{A}^{\dagger}$; and (c) $(\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1}$,

Setting a Price A company that manufactures game consoles is attempting to determine the best price for their newest unit. If the price is too high, no one will buy the consoles, and the company will have no revenue. If the price is too low, the cost to manufacture the units will be greater than the selling price. After market research, the company determines that the revenue can be modeled by the function $r(x)=-2 x^{2}+42 x$, where $r(x)$ is measured in thousands of dollars and $x$ represents the number of units sold in thousands. Part B An executive with the company believes that it is possible for revenue from console sales to be $\$ 250,000$. Do you agree or disagree? Justify your response.

For the points \[ A(0,-3,6), B(-2,-8,-8) \text { and } C(5,8,-1) \] find the position vector $\mathbf{t}$ of the point $T$ on the straight line through $A$ and $B$ such that \[ \overrightarrow{A T}=-4 \overrightarrow{T B} \] Enter your position vector $\mathbf{t}$ in the box below using Maple notation. For example, for the vector $\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$ enter $<1,2,3>$. \[ \mathbf{t}= \] Q I dont understand how to do this question.

The functions $f(x)=-2.9 x+287$ and $g(x)=0.01 x^{2}-4.9 x+370$ model the chance, as a percent, that a 60 -year-old will survive to age $x$. The bar graph to the right shows the chances of 60 -year-olds surviving to various ages. a. Find and interpret $f(80)$. b. Find and interpret $g(80)$. c. Which function serves as a better model for the chance of surviving to age 80 ? Chances of 60-Year-Olds Surviving to Various Ages a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answers.) A. $f(80)=\square$, which means that the chance of a 60 -year-old surviving to the age of $\square$ is $80 \%$. B. $f(80)=\square$, which means that the chance of a 60 -year-old surviving to the age of 80 is $\square \%$.

-4) On the island of Foo, the natives use a counting system based on colored pebbles. In this system: A red pebble is equal to 3 yellow pebbles A blue pebble is equal to a red pebble and 2 yellow pebbles A green pebble is equal to 3 red pebbles Using the smallest number of pebbles possible and without using blue pebbles, what is the value in pebbles of 2 blue pebbles 2 red pebbles 2 red pebbles and 1 yellow pebble 3 red pebbles and 1 yellow pebble 1 green pebble and 1 yellow pebble 2 green pebbles 5) All of the employees at the AcmeSoft company are

Factogon In the blank rectangles, write an algebraic expression such that any one of the given trinomials is the product of the two expressions found on both ends of the segment.

Module 14 Discussion Question Refer back to the Road Trip Example in the presentation slides. Assume that after sitting at McDonald s for 1 hour, 1 got back on the highway and again made good time averaging $78 \mathrm{mph}$ for the next 1.5 hours followed by 45 minutes averaging $35 \mathrm{mph}$ before stopping for the night. Create the piecewise function that models the complete road trip for that day. Hint start with the piecewise function defined in the presentation and add these next 3 pieces of the function. Do not just write out the function, instead explain the process/reasoning you used to create the rules and conditions that make up these last 3 pieces of the function. Be sure that your work is organized and legible and easy to follow.

We proved in the text that if $T: R^{n} \rightarrow R^{m}$ is a matrix transformation, then $T(\mathbf{0})=\mathbf{0}$ . Show that the converse of this result is false by finding a mapping $T: R^{n} \rightarrow R^{m}$ that is not a matrix transformation but for which $T(\mathbf{0})=\mathbf{0}$ . We proved in the text that if $T: R^{n} \rightarrow R^{m}$ is a matrix transformation, then $T(\mathbf{0})=\mathbf{0}$. Show that the converse of this result is false by finding a mapping $T: R^{n} \rightarrow R^{m}$ that is not a matrix transformation but for which $T(\mathbf{0})=\mathbf{0}$.

In Exercises 5-6 an elementary matrix E and a matrix A are given. Identify the row operation corresponding to E and verify that the product E A results from applying the row operation to A .>(a) $E=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right], \quad A=\left[\begin{array}{rrrr}-1 & -2 & 5 & -1 \\ 3 & -6 & -6 & -6\end{array}\right]$ (b) $E=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -3 & 1\end{array}\right], \quad A=\left[\begin{array}{rrrrr}2 & -1 & 0 & -4 & -4 \\ 1 & -3 & -1 & 5 & 3 \\ 2 & 0 & 1 & 3 & -1\end{array}\right]$ (c) $E=\left[\begin{array}{lll}1 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right], \quad A=\left[\begin{array}{ll}1 & 4 \\ 2 & 5 \\ 3 & 6\end{array}\right]$ In Exercises 5-6 an elementary matrix $E$ and a matrix $A$ are given. Identify the row operation corresponding to $E$ and verify that the product $E A$ results from applying the row operation to $A$. (a) $E=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right], \quad A=\left[\begin{array}{rrrr}-1 & -2 & 5 & -1 \\ 3 & -6 & -6 & -6\end{array}\right]$ (b) $E=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -3 & 1\end{array}\right], \quad A=\left[\begin{array}{rrrrr}2 & -1 & 0 & -4 & -4 \\ 1 & -3 & -1 & 5 & 3 \\ 2 & 0 & 1 & 3 & -1\end{array}\right]$ (c) $E=\left[\begin{array}{lll}1 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right], \quad A=\left[\begin{array}{ll}1 & 4 \\ 2 & 5 \\ 3 & 6\end{array}\right]$

In Exercises 23-24, solve the matrix equation for $a, b, c$ , and d .> $\left[\begin{array}{cc}a-b & b+a \\ 3 d+c & 2 d-c\end{array}\right]=\left[\begin{array}{ll}8 & 1 \\ 7 & 6\end{array}\right]$ In Exercises 23-24, solve the matrix equation for $a, b, c$, and $d$. $\left[\begin{array}{cc}a-b & b+a \\ 3 d+c & 2 d-c\end{array}\right]=\left[\begin{array}{cc}8 & 1 \\ 7 & 6\end{array}\right]$