Linear Algebra Questions and Answers

2 9 Given that the set 1 1 x x p x is a basis for P2 which of the following is a possible value for p x A 0 C 1 D 2 x x B 1 x 7

The solid below is made from 1 inch cubes Find the total surface area of the solid Be sure to include the correct unit in your answer 1 inch 0 in X in in S

21 On a large residential college campus one student returns from vacation with a contagious and long lasting flu virus The spread of the virus is modeled by 6000 1 5999 e 0 8t y t 0 where y is the total number of students infected after t days a What is the limiting number of students who will become infected b What are the two horizontal asymptotes of the graph of this logistic function c Find how many days it will take for 4000 students to become infected Find an expression for the exact answer and simplify as much as you can without a calculator

Suppose your credit card has a balance of 5 100 and an annual interest rate of 14 You decide to pay off the balance over three years If there are no further purchases charged to the card a How much must you pay each month b How much total interest will you pay Now suppose decide to pay off the balance over one year rather than three c How much more must you pay each month d How much less will you pay in total interest OA a 174 b 396 c 284 more per month d 768 less in total interest OC a 181 b 480 c 284 more per month d 936 less in total interest www O B a 181 Use PMT OD a 174 b 1 416 c 284 more per month d 936 less in total interest 1 to determine the payment amount Round to the nearest dollar b 1 164 c 284 more per month d 768 less in total interest PA

Suppose that there are two types of tickets to a show advance and same day Advance tickets cost 20 and same day tickets cost 15 For one performance there were 50 tickets sold in all and the total amount paid for them was 850 How many tickets of each type were sold Number of advance tickets sold Number of same day tickets sold X

The principal P 400 is borrowed at simple interest rate r 6 25 for a period of time t 3 months Find the sin interest owed for the use of the money Assume 360 days in a year and round to the nearest cent A 6 82 OB 4 25 OC 6 25 OD 75 00 THE

Michael is driving to Denver Suppose that the remaining distance to drive in miles is a linear function of driving time in minutes When graphed the function gives a line with a slope of 0 75 See the figure below Michael has 51 miles remaining after 41 minutes of driving How many miles will be remaining after 53 minutes of driving Remaining distance in miles 51 41 Driving time in minutes miles X

A bank offers a CD that pays a simple interest rate of 5 How much must you put in this CD now in order to have 6 900 to replace all the windows in your house in 3 years A 6 000 OB 1 800 OC 900 D 90

3 The height h in meters of a car above the ground as a Ferris wheel turn can be t modelled using the function h BO sin CD where t is the time in seconds a Determine the average rate of change of h over each time interval round to three decimal places i ii iii 5s to 10 s 9 9 s to 10 s 9 99 s to 10 s b Estimate the value for the instantaneous rate of change of h at t 10 s c What physical quantity does this instantaneous rate of change represent d Would you expect the instantaneous rate of change of h to be the same at t 15 s Justify your answer

At Greg s auto shop it takes him 9 minutes to do an oil change and 15 minutes to do a tire change Let x be the number of oil changes he does Let y be the number of tire changes he does Using the values and variables given write an inequality describing how many oil changes and tire changes Greg can do in at most 3 hours 180 minutes

y year 1950 6 80 1960 7 05 1970 8 89 1980 8 10 1990 6 59 m minimum wage e the table shown to estimate the minimum wage in the year 1967 A 7 23 B 8 51 c 8 67 D 8 34

Question Transform orthogonal set 2 1 0 to

mixing system consists of two tanks Tank 1 has a volume of 1000 m Thank 2 is half size of Tank 1 Initially Tank 1 is full of salt brine of 1 concentration and Tank 2 is full of pure water Circulation of well mixed solution between the tanks starts at time t 0 at a rate of 1 m s What is the concentration of the brine in Tank 1 at 1 minute Round your answer to 3 decimal places For example if the concentration is 0 5318 Enter 0 532 Answer

2 5 points If the graph of y f x is shown below sketch the graph of y sure to label inflection points clearly 4 2 4 2 2 4

7 From point A on the ground the angle of elevation to the top of a Redwood tree is 32 degrees From point B on the ground 150 yards closer to the tree the angle of elevation to the top of the tree is 41 degrees a Sketch a picture of this situation b Find the height of the Redwood tree to the nearest whole yard

Write the statement in symbolic form Let p and q represent the following statements p The teacher is at the blackboard q The students are at their desks The students are not at their desks or the teacher is not at the blackboard The statement in symbolic form is q Vp

If after row operations a system of linear equations takes the form 4 7x 1 7y 5 5z 3 8 0x Oy Oz 0 Then what is the correct form describing the family of infinite solutions and what does it represent geometrically O a This is a trick question A line of zeroes in a system of linear equations means there are no solutions O b The family of infinite solutions will take the form 0 81 0 36 s 1 17 t s t t and s going from negative infinity to positive infinity This scenario represents two equations representing the same plane O c The family of infinite solutions will take the form 3 80 1 70 s 5 50 t s t t and is going from negative infinity to positive infinity This scenario represents two equations representing the same plane O d The family of infinite solutions will take the form 0 81 0 36 s 1 17 s t this scenario represents two equations representing the same line Oe The family of infinite solutions will take the form 0 81 0 36 s 1 17 t s t t and s going from negative infinity to positive infinity This scenario represents two equations representing the same line Of The family of infinite solutions will take the form 0 81 0 36 t 1 17 0 1 going from negative infinity to positive infinity This scenario represents two equations representing the same line

Find the maximum value of the function z 2x 3y subject to the following constraints y20 x25 y 4 2x y 22 z 0 30 2

Use Green s Theorem to evaluate the following line integral for dy dy g dx where f g 10x 11y and C is the upper half of the unit circle and the line segment 1 x 1 oriented clockwise C fr dy dy g dx

You have earned 0 point s out of 0 point s thus far Given that sin 2x Z y if x dx 0 25 find the differential Round your answer to three decimal places Your answer

Use a graphing utility to construct a table of values for the function Round your answers to three decimal places f x 4e 1 1x X f x 2 Sketch the graph of the function 1 0 1 f x 2

Describe the transformation of the graph of f that yields the graph of g f x 0 3 g x 0 3 9 The graph of g can be obtained by reflecting the graph of f in the Select and shifting the resulting graph unit s

Solve the system graphically or algebraically If there is no solution enter NO SOLUTION x x y X x y x y 8 0 0 8 64 y 8 smaller x value larger x value Explain your choice of method

Find the dimensions of the rectangle meeting the specified conditions The perimeter is 24 4 millimeters and the length is 2 4 times the width Round your answers to one decimal place width length mm mm

Find the Minimal Spanning Tree of the following graph 7 a 1 8 8 O a b c d f e a b a b c d f g 6 10 9 f e 2 O Minimal spanning tree is not possible O a b c d e f g 3 5 C 4 7 d

Put the following equation of a line into slope-intercept form, simplifying all fractions. 4x- 20y = -160

Problem 4 (5 points) Solve the system of linear equations by Gaussian elimination. x - 2y + 3z = 4 2x + y + 4z = 3 -3x+4y-z = -2

An employee earns a salary of $750 per week and a 4% commission on all sales. Determine the employee's total pay if her sales for the week total $18,100. a. $1,474 b. 822.40 c. $750 d. $724

Compute the total commission for the following commission payment plan. Graduated Commission Rates Sales Commission 1% on sales to $125,000 $224,000 2% on sales above $125,000 a. $3,230 b. $2,240 c. $4,480 d. $3,200

What was the cost of each item? Solve on paper. Then check your work on Zearn. The burger cost $ The souvenir cost $ The pass cost $

Let f:R to R be a continuous function at x0,x1,..,xn. Alberto states: Since fis continuous, then there exists a unique polynomial P of degree n such that P(xi)=f(xi), for i=0,1,...,n. If you agree with Alberto, prove it. Otherwise an example

Find the projection of u onto v. u = (6,4) = (6, -9) V= projvu = Write u as the sum of two orthogonal vectors, one of which is projvu. U= projvu +

A certain cut of beef costs $7.59 per pound, and a similar cut of pork costs $5.19 per pound. What is the total cost of 3.25 pounds of the beef and 3.75 pounds of the pork? (Round to the nearest cent.) a. $56.45 total b. $44.13 total c. $41.00 total d. $45.33 total

Solve and check the linear equation. x/3=5+ x-35/5 What is the solution? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Simplify your answer.) B. The solution set is {x|x is a real number}. C. The solution set is Ø.

Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your answer as a function of t.) L^-1 ( 7/ s^2+64)

Let E be an algebraic extension of F and o: E→ E be an embedding such that o| F = Id. Then, σ is an automorphism of E.

For u, v € R³, which of the following is (are) inner product(s)? (A) (u, v) = U₁V₁ + ZU₂V₂ + 5u3V3 (B) (u, v) = U₁V₁ - U₂V₂ + ZU3V3 (C) (u, v) = u²v₁ + U₂V² + U3V3 (D) (u, v) = U₁V₂ + U2V3 + U3V1 (E) (u, v) = U₁U₂Uz + V₁ V₂ V3 Which of the following is (are) linear transformation(s)? (A) f(x) = sin x + cos x (B) f(x) = ex(x² + x). (C) T(x, y) = (x + y, x - y). (D) T(x, y, z) = (x − 2y, 0, x + 3y - z). (E) T(x, y) = (x − 1, y + 1, x + z)

Let f(x) = F[x] be a nonconstant polynomial and a be an element in any field extension E of F. Then, a is a multiple root of f(x) if and only if f(a) = 0 = f'(a); that is, a is a root of both f'(x) and f'(x).

Find the component form of v, where u = 4i- j and w = i + 4j. v=u - 4w Sketch the vector operation geometrically.

Find the component form of the sum of u and v with direction angles θu and θv. Magnitude ||u|| = 6 ||v|| = 6 θu = 60° θv = 90°

Find the component form and magnitude of the vector v. Terminal Point (17, -21) Initial Point (-1,2) V= ||v|| =

Find the equation of the line using the given information. The slope equals zero and it passes through the point (x, y) = (1,-4).

Let F = GF(p^n) and m be a positive integer. Then there exists a field extensions E of F such that [E: F] = m and any two such extensions are isomorphic.

Let P₂ denote the vector space of all polynomials in the variable x of degree less than or equal to 2. Let C = {3, 2-3x,-2-2x + x²} be an ordered basis for P2. a. Write 5 + x + x² as a linear combination of elements from the basis C. 5+x+x² = b. Let [q]c denote the coordinate representation of q relative to the basis C. Find the coordinate vector representation for 5 + x + x² relative to the basis C. Your answer should be vector of the general form <1,2,3>. [5+ x + x²]c=

Suppose V₁, V₂, V₃ is an orthogonal set of vectors in R⁵ with V₁ V₁ = 35, V₂ V₂ = 24, V₃ V₃ = 36. Let w be a vector in Span(V₁, V₂, V₃) such that w. v₁ = -35, w V₂ = -48, w V₃=108. then w= __ v₁+___v₂+___v₃

Find dimension of the linear span of vectors (4; 4;-1; 3;-1); (-1;-2; 2; 0; 4); and (-8;-8; 2;-6; 2). (A) 2 (B) 3

Let F be any field and n be a positive integer. Let M be the set of all polynomials over F of degree less than n. Then, prove that M is a finitely generated F-module and exhibit two distinct sets of generators of M, each with n elements.

Let F be any field and consider the vector spaces F³ and F² over F. Define a F³ → F² by a(a, b, c) = (a + b + c, b + c) Then, prove that a is a linear transformation and determine the matrix of a with respect to the standard basis of F³ and F².

Consider the ring Z of integers. Then, Z is a mod- ule over itself. For each positive integer n, prove that the Z-module Z has an n-element generating set X such that no proper subset of X generates Z.

Evaluate the expression when c = 63 and d = 17. d-c/7