Assignment linearalgebrahw1 due 10/15/2012 at 02:32pm EDT

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1 mustafa zeki Assignment linearalgebrahw1 due 10/15/2012 at 02:32pm EDT math (1 pt) Library/Rochester/setLinearAlgebra24SingularValues- /ur la 24 7.pg -1 7 Let A = A singular value decomposition of A is as follows: A = Find the least-squares solution of the linear system -1 A b, where b = x1 =, x2 =. 2. (1 pt) Library/Rochester/setLinearAlgebra3Matrices/ur la 3 33.pg Find a non-zero, two-by-two matrix such that: = \(\displaystyle\left.\beginarray}cc} \mbox } &\mbox } \cr \mbox } &\mbox } \cr 3. (1 pt) Library/Rochester/setLinearAlgebra3Matrices/ur Ch1 3 4.pg 5x + 7y 8z = 3 3x + 8y 5z = 3 4x + 1y 5z = 7 Write the above system of equations in matrix form: x y = z \(\displaystyle\left.\beginarray}ccc} \mbox5} &\mbox7} &\mbox-8} \cr \mbox3} &\mbox8} &\mbox-5} \cr \mbox4} &\mbox1} &\mbox-5} \cr \(\displaystyle\left.\beginarray}c} \mbox3} \cr \mbox-3} \cr \mbox7} \cr (1 pt) Library/Rochester/setLinearAlgebra3Matrices/ur la 3 25.pg x 5 If A =, determine the values of x and y for which y -6 A 2 = A., (1 pt) Library/Rochester/setLinearAlgebra3Matrices/ur la 3 15.pg Find a and b such that = a 3 +b a = b = (1 pt) Library/Rochester/setLinearAlgebra3Matrices/ur la 3 34.pg Find a non-zero, two-by-two matrix such that: 0 0 A 2 = A = 0 0 \(\displaystyle\left.\beginarray}cc} \mbox1} &\mbox1} \cr \mbox-1} &\mbox-1} \cr 7. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 16.pg Solve the system x +y= 6 5x 3y= 6 11x 5y= (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 12.pg Determine the value of k for which the system has no solutions. k = x +y+5z= 1 x +2y 3z= 1 6x+13y+kz= 9

2 6 9. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 3.pg Solve the system using substitution x 3y= x 8y= (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 4.pg Solve the system using substitution 8x+5y= 43 7x+4y= (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 5.pg Solve the system using elimination z = 2x+3y+5z= 35 3x+2y 4z= 19 6x 5y+2z= (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 17.pg Solve the system x 1 = x 2 = x 3 = x 4 = 7 x 1 +2x 3 +2x 4 = 26 x 2 4x 3 3x 4 = 38 2x 1 3x 2 +18x 3 +13x 4 = 180 x 2 +4x 3 +7x 4 = (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 15.pg Solve the system x 1 x 2 = x 3 x 1 x 2 +3x 3 = 6 4x 1 3x 2 +5x 3 = 3 3x 1 +12x 3 = 45 + s. \(\displaystyle\left.\beginarray}c} \mbox15} \cr \mbox21} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox4} \cr \mbox7} \cr \mbox1} \cr 14. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 14.pg Solve the system x1 +x 2 +4x 3 = 8 x 1 x 2 = x 3 + 3x 1 +2x 2 3x 3 = 6 s. \(\displaystyle\left.\beginarray}c} \mbox10} \cr \mbox-18} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox11} \cr \mbox-15} \cr \mbox1} \cr 15. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 4b.pg Solve the system using matrices (row operations) x+5y= 69 7x 5y= 51 2

3 16. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 6.pg For each system, determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions. 1. 2x+3y= x+9y= 59 A. Unique solution: 59, 20 B. No solutions C. Unique solution: 0, 0 D. Infinitely many solutions E. Unique solution: 20, 59 F. None of the above 2x+2y=10 6x+6y=30 A. Unique solution: 0, 0 B. Infinitely many solutions C. Unique solution: 10, 30 D. No solutions E. Unique solution: 5, 0 F. None of the above 9x+7y=0 6x+3y=0 A. Unique solution: 2, 3 B. No solutions C. Unique solution: 3, 9 D. Infinitely many solutions E. Unique solution: 0, 0 F. None of the above 5x+4y= 0 6x 7y=11 A. No solutions B. Unique solution: 4, 5 C. Unique solution: 5, 4 D. Unique solution: 0, 0 E. Infinitely many solutions F. None of the above B B E B (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 4a.pg Solve the system using elimination 2x 7y= x+8y= (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 1.pg Perform one step of row reduction, in order to calculate the values for x and y by back substitution. Then calculate the values for x and for y. Also calculate the determinant of the original matrix. You can let webwork do much of the calculation for you if you want (e.g. enter 45-(56/76)(-3) instead of calculating the value out). You can also use the preview feature in order to make sure that you have used the correct syntax in entering the answer. Note since the determinant is unchanged by row reduction it will be easier to calculate the determinant of the row reduced matrix x 10-8 y 2 20 x y 0 det = -2 = 4 = (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 21.pg x 1 x 2 The dot product of two vectors. x n and y 1 y 2. y n in R n is defined by x x 1 y 1 + x 2 y x n y n. The vectors x and y are called perpendicular if x 0. Then any vector in R 3 perpendicular to -2-2 can be written -3 in the form s + t. \(\displaystyle\left.\beginarray}c} \mbox-2} \cr

4 \mbox2} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox-3} \cr \mbox0} \cr \mbox2} \cr 20. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 4c.pg Solve the system by using Cramer s Rule. 4x+7y= 11 7x+2y= (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 22.pg The reduced row-echelon forms of the augmented matrices of four systems are given below. How many solutions does each system have? A. Infinitely many solutions B. No solutions C. Unique solution D. None of the above A. Unique solution B. No solutions C. Infinitely many solutions D. None of the above A. Infinitely many solutions B. No solutions C. Unique solution D. None of the above A. No solutions B. Infinitely many solutions C. Unique solution D. None of the above B C A C (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 23.pg Write the system 2y 3z= 2 7x x 4y 6z= 9 in matrix form x y z = 23. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 20.pg Solve the system x 1 4x 2 +2x 3 4x 5 +5x 6 =4 x 4 5x 5 +5x 6 =5 x 1 4x 2 6x 5 9x 6 =4 x 1 x 2 x 3 x 4 x 5 x 6 + = u. + s + \(\displaystyle\left.\beginarray}c} \mbox4} \cr \mbox0} \cr \mbox0} \cr \mbox-5} \cr \mbox0} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox4} \cr \mbox1} \cr \mbox0} \cr \mbox0} \cr \mbox0} \cr \mbox0} \cr t

5 ,\(\displaystyle\left.\beginarray}c} \mbox6} \cr \mbox0} \cr \mbox-1} \cr \mbox-5} \cr \mbox1} \cr \mbox0} \cr x y =,\(\displaystyle\left.\beginarray}c} z \mbox9} \cr \mbox0} \cr \mbox-7} \cr \mbox5} \cr \mbox0} \cr \mbox1} \cr 24. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 7.pg Write the augmented matrix of the system x +3z=3 8x 2y z=4 47x (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 5a.pg Solve the system using matrices (row operations) z = -5 2x 4y+5z= 13 3x+2y+4z= 10 2x 3y 2z= (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 10.pg Solve the equation 8x + 7y + 5z = s + t. \(\displaystyle\left.\beginarray}c} \mbox1.25} \cr \mbox0} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox0.875} \cr \mbox1} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox0.625} \cr \mbox0} \cr \mbox1} \cr 27. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 19.pg Solve the system x 1 x 2 x 3 = x 4 4x 1 5x 2 +2x 3 +4x 4 = 0 x 1 +x 2 +3x 3 +3x 4 = 2 3x 1 4x 2 +5x 3 +7x 4 = 2 3x 1 3x 2 9x 3 9x 4 = 6 + s + t. \(\displaystyle\left.\beginarray}c} \mbox-10} \cr \mbox-8} \cr \mbox0} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox17} \cr \mbox14} \cr \mbox1} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox19} \cr \mbox16} \cr \mbox0} \cr \mbox1} \cr 28. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 9.pg Determine the value of h such that the matrix is the augmented matrix of a linear system with infinitely many solutions h 2 6

6 h = (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 11.pg Solve the system (a*-6-5*b)/1 (4*b-a*-5)/1 4x+5y=a 5x 6y=b 30. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 13.pg Solve the system 2x1 +x 2 = 3 6x 1 3x 2 = 9 = + s. x1 x 2 \(\displaystyle\left.\beginarray}c} \mbox1.5} \cr \mbox0} \cr matrix.,\(\displaystyle\left.\beginarray}c} \mbox-0.5} \cr \mbox1} \cr 31. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 8.pg Determine the value of h such that the matrix is the augmented matrix of a consistent linear system. h = h 8 \(\displaystyle\left.\beginarray}c} \mbox5} \cr \mbox-3} \cr \mbox-2} \cr \mbox0} \cr,\(\displaystyle\left.\beginarray}c} \mbox-1} \cr \mbox1} \cr \mbox-1} \cr \mbox1} \cr 33. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 2.pg Perform one step of row reduction, in order to calculate the values for x and y by back substitution. Then calculate the values for x and for y. Also calculate the determinant of the original matrix. You can let webwork do much of the calculation for you if you want (e.g. enter 45-(56/76)(-3) instead of calculating the value out). You can also use the preview feature in order to make sure that you have used the correct syntax in entering the answer. This problem has rather difficult complex calculations. Note since the determinant is unchanged by row reduction it will be easier to calculate the determinant of the row reduced 1+2i -2-4i x -2+2i 4-2i y 1+2i -2-4i 0 det = i -4+2i -1+3i +2i -4+2i -1-7i = 4-2i x y = -1-7i 32. (1 pt) Library/Rochester/setLinearAlgebra1Systems/ur la 1 18.pg Solve the system x 1 +x 2 = 2 x 2 +x 3 = 5 x 3 +x 4 = 2 x 1 +x 4 = 5 x 1 x 2 x 3 = x 4 + s (1 pt) Library/Rochester/setLinearAlgebra2SystemsApplications- /ur la 2 2.pg Find the quadratic polynomial whose graph goes through the points ( 2,6), (0,4), and (2,18). f (x) = x 2 + x+

7 35. (1 pt) Library/Rochester/setLinearAlgebra2SystemsApplications- /ur la 2 7.pg Consider a two-commodity market. When the unit prices of the products are P 1 and P 2, the quantities demanded, D 1 and D 2, and the quantities supplied, S 1 and S 2, are given by D 1 = 55 3P 1 + P 2 D 2 = 163+ P 1 3P 2 S 1 = 28+2P 1 S 2 = 22 +2P 2 (a) What is the relationship between the two commodities? Do they compete, as do Volvos and BMWs, or do they complement one another, as do shirts and ties? (type in compete or complement ) (b) Find the equilibrium prices (i.e. the prices for which supply equals demand), for both products. P 1 = P 2 = compete (1 pt) Library/Rochester/setLinearAlgebra2SystemsApplications- /ur la 2 6.pg In a grid of wires, the temperature at exterior mesh poins is maintained at constant values as shown in the figure. When the grid is in thermal equilibrium, the temperature at each interior mesh point is the average of the temperatures at the four adjacent points. For instance, T 1 = T 2 + T Find the temperatures T 1, T 2, T 3, T 4, when the grid is in thermal equilibrium. T 3 = T 4 = (1 pt) Library/Rochester/setLinearAlgebra2SystemsApplications- /ur la 2 1.pg Tonya and Steve are sister and brother. Tonya has twice as many brothers as sisters, and Steve has as many brothers as sisters. How many girls and boys are there in this family? Answer: girls and boys. 38. (1 pt) Library/Rochester/setLinearAlgebra2SystemsApplications- /ur la 2 8.pg A dietitian is planning a meal that supplies certain quantities of vitamin C, calcium, and magnesium. Three foods will be used, their quantities measured in milligrams. The nutrients supplied by one unit of each food and the dietary requirements are given in the table below. Nutrient Food 1 Food 2 Food 3 Total Required (mg) Vitamin C Calcium Magnesium Write the augmented matrix for this problem. What quantity (in units) of Food 1 is necesary to meet the dietary requirements? T 1 = T 2 = 7 What quantity (in units) of Food 2 is necesary to meet the dietary requirements? What quantity (in units) of Food 3 is necesary to meet the dietary requirements?

8 (1 pt) Library/Rochester/setLinearAlgebra2SystemsApplications- /ur la 2 5.pg Consider the chemical reaction an 2 H 4 + bn 2 O 4 cn 2 + dh 2 O, where a, b, c, and d are unknown positive integers. The reaction mush be balanced; that is, the number of atoms of each element must be the same before and after the reaction. For example, because the number of oxygen atoms must remain the same, 4b = d. While there are many possible choices for a, b, c, and d that balance the reaction, it is customary to use the smallest possible integers. Balance this reaction. a = b = c = d = 40. (1 pt) Library/Rochester/setLinearAlgebra2SystemsApplications- /ur la 2 3.pg Find the polynomial of degree 4 whose graph goes through the points ( 2, 54), ( 1, 3), (0,2), (2, 6), and (3, 139). f (x) = x 4 + x 3 + x 2 + x (1 pt) Library/Rochester/setLinearAlgebra2SystemsApplications- /ur la 2 4.pg Find the cubic polynomial f (x) such that f (1) = 3, f (1) = 6, f (1) = 16, and f (1) = 12. f (x) = x 3 + x 2 + x (1 pt) Library/Rochester/setAlgebra34Matrices/sw7 4 5.pg Given the matrices B =, C =, find 3B + 2C. Write 3B + 2C as a11 a 3B + 2C = 12 a 13. a 21 a 22 a 23 Input your answer below: a 11 = a 12 = a 13 = a 21 = a 22 = a 23 = (1 pt) Library/Rochester/setAlgebra34Matrices/sw7 4 1.pg Given the matrices B =, C =, find B +C. Write B +C as a11 a B +C = 12 a 13. a 21 a 22 a 23 Input your answer below: a 11 = a 12 = a 13 = a 21 = a 22 = a 23 = (1 pt) Library/Rochester/setAlgebra34Matrices/scalarmult3.pg If A = and B = Then 2A + B = and A T = 8

9 (1 pt) Library/Rochester/setAlgebra34Matrices/sw7 4 3.pg Given the matrices B =, C =, find C B. Write C B as a11 a C B = 12 a 13. a 21 a 22 a 23 Input your answer below: a 11 = a 12 = a 13 = a 21 = a 22 = a 23 = (1 pt) Library/Rochester/setAlgebra34Matrices/scalarmult3a.pg If A = and B = Then 2A 2B = and 4A T = (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 4.pg Find the least-squares solution x of the system x = (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 7.pg Fit a trigonometric function of the form f (t) = c 0 + c 1 sin(t) + c 2 cos(t) to the data points (0, 2), ( π 2,5), (π, 12), ( 3π 2, 7), using least squares. c 0 =, c 1 =, c 2 = (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 6.pg Fit a quadratic function of the form f (t) = c 0 + c 1 t + c 2 t 2 to the data points (0, 7), (1, 13), (2, 11), (3, 21), using least squares. c 0 =, c 1 =, c 2 =

10 50. (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 3.pg Find the least-squares solution x of the system x =. 51. (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 9.pg The table below lists the height h (in cm), the age a (in years), the gender g (1= Male, 0= Female ), and the weight w (in kg) of some college students. Height Age Gender Weight We wish to fit a linear function of the form w(t) = c 0 + c 1 h + c 2 a + c 3 g which predicts the weight from the rest of the data. Find the best approximation of this function, using least squares. c 0 =, c 1 =, c 2 =, c 3 = (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 8.pg Let S(t) be the number of daylight hours on the tth day of the year in Manley Hot Springs. We are given the following data for S(t): Day t S(t) January March May July We wish to fit a trigonometric function of the form ( ) ( ) 2π 2π f (t) = a + bsin 365 t + ccos 365 t to these data. Find the best approximation of this form, using least squares. a =, 10 b =, c = (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la pg During the summer months Terry makes and sells necklaces on the beach. Terry notices that if he lowers the price, he can sell more necklaces, and if he raises the price than he sells fewer necklaces. The table below shows how the number n of necklaces sold in one day depends on the price p (in dollars). Price Number of necklaces sold (a) Find a linear function of the form n = c 0 + c 1 p that best fits these data, using least squares. c 0 =, c 1 =. (b) Find the revenue (number of items sold times the price of each item) as a function of price p. R =. (c) If the material for each necklace costs Terry 5 dollars, find the profit (revenue minus cost of the material) as a function of price p. P =. (d) Finally, find the price that will maximize the profit. p = *p *p*p *(p - 5) *p*(p-5) (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 5.pg Fit a linear function of the form f (t) = c 0 +c 1 t to the data points ( 9,60), (0,0), (9, 66), using least squares. c 0 =, c 1 =. -7

11 55. (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 2.pg Find the least-squares solution x of the system x = (1 pt) Library/Rochester/setLinearAlgebra20LeastSquares- /ur la 20 1.pg Find the least-squares solution x of the system x = (1 pt) Library/maCalcDB/setLinearAlgebra1Systems/ur la 1 3.pg Solve the system using substitution x 4y= x 14y=4 58. (1 pt) Library/maCalcDB/setLinearAlgebra1Systems/ur la 1 4.pg Solve the system using substitution 4x+3y= 42 9x+5y= (1 pt) Library/maCalcDB/setLinearAlgebra1Systems/ur la 1 5.pg Solve the system using elimination z = 2x 2y 5z= 21 5x 2y 2z= 26 5x 3y+4z= (1 pt) Library/maCalcDB/setLinearAlgebra1Systems- /ur la 1 4a.pg Solve the system using elimination 8x+5y= x 4y= (1 pt) Library/maCalcDB/setLinearAlgebra1Systems/ur la 1 7.pg Write the augmented matrix of the system x 69y +9z=63 18y 6z= 0 91x +93z= (1 pt) Library/TCNJ/TCNJ LinearSystems/problem2.pg Determine whether the following system has no solution, an infinite number of solutions or a unique solution.? 1. 10x + 10y 25z = 0 6x 6y + 15z = 0? 2. 10x + 10y 25z = 0 6x 6y + 15z = 1? 3. 3x 4y + 3z = 1 3x + 3y + 7z = 7 Infinite Solutions No Solution Infinite Solutions

12 63. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem18.pg Determine all values of h and k for which the system 9x + 5 h 6x + k 7 has no solution. k = h (1 pt) Library/TCNJ/TCNJ LinearSystems/problem4.pg Give a geometric description of the following system of equations? 1.? 2.? 3. 2x + 4y 6z = 12 3x 6y + 9z = 18 2x + 4y 6z = 12 x + 5y 9z = 1 2x + 4y 6z = 12 3x 6y + 9z = 16 Two planes that are the same Two planes intersecting in a line Two parallel planes 65. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem7.pg Solve the system. x 4 1 2x (1 pt) Library/TCNJ/TCNJ LinearSystems/problem15.pg Solve the system: 4x 3 a 3x 2 b (a*-2--3*b)/1 (4*b-a*3)/ (1 pt) Library/TCNJ/TCNJ LinearSystems/problem1.pg Determine whether the following system has no solution, an infinite number of solutions or a unique solution. 5x +5 3? 1. 4x x x +5 3? 2. 4x x x +20 5? 3. 8x x Unique Solution No Solution Infinite Solutions 68. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem19.pg The system 6x 30y 35z = 0 8x + 39y + 46z = 0 5x + 25y + 30z = 0 has the solution,, z =. 69. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem14.pg Solve the system using elimination z = x 2y+3z= 5 3x+2y+4z= 18 6x 5y+4z= (1 pt) Library/TCNJ/TCNJ LinearSystems/problem13.pg Determine the value of k for which the system has no solutions. k = x +y+5z= 2 x +2y 3z= 2 5x+12y+kz=11

13 1 71. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem8.pg The system 3 x 15 y z = 4 9 x + 44 y z = 1 5x has the solution,, z = (1 pt) Library/TCNJ/TCNJ LinearSystems/problem16.pg Determine whether the following systems have no solution, an infinite number of solutions or a unique solution.? 1.? 2.? 3. 5x x x x x x x x x Unique Solution No Solution Infinite Solutions 73. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following systems of equations? 1.? 2.? 3. x x x x 8 4 8x x 12 6 x x x Three non-parallel lines with no common intersection Three identical lines Three lines intersecting at a single point 74. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem11.pg Give a geometric description of the following systems of equations.? 1.? 2.? 3. 2x x x x x 5 8 4x Two parallel lines Two lines that are the same Two lines intersecting in a point 75. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem12.pg x 2y + 9z = 3 x 7y + 9z = 13 3x 11y + 27z = k In order for the above system of equations to be a consistent system, then k must be equal to (1 pt) Library/TCNJ/TCNJ LinearSystems/problem5.pg Determine whether the following system has no solution, an infinite number of solutions or a unique solution.? 1.? 2.? 3. 3x x 3 1 6x x x x Unique Solution Infinite Solutions No Solution 77. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem9.pg The system 5x x has the solution:, 13

14 78. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem10.pg Give a geometric description of the following system of equations.? 1.? 2.? 3. 2x x x x x x Two lines intersecting in a point Two parallel lines Two lines that are the same 79. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem6.pg Give a geometric description of the following systems of equations x + 4y + 16z = 1? 4. x 3y 12z = 2 4x + 16y + 60z = 6 Infinite Solutions Infinite Solutions No Solution Unique Solution 81. (1 pt) Library/TCNJ/TCNJ SolutionSetsLinearSystems- /problem7.pg Let A = Describe all solutions of A 0. x 2 +x 4 +x 6? 1.? 2.? 3.? 4. x + 3y + 8z = 5 4x 11y 26z = 4 3x + 9y + 21z = 5 7x + 7y + 5z = 1 4x 3y + z = 3 15x 13y 3z = 12 7x + 7y + 5z = 1 4x 3y + z = 3 15x 13y 3z = 7 8x + 10y 4z = 4 20x 25y + 10z = x + 35 y 14 z = 14 Three planes intersecting at a point Three planes with no common intersection Three planes intersecting in a line Three identical planes 80. (1 pt) Library/TCNJ/TCNJ LinearSystems/problem17.pg Determine whether the following system has no solution, an infinite number of solutions or a unique solution.? 1.? 2.? 3. 3 x + y + 5 z = 3 3x 5y + 4z = 3 9x 15y 17z = 21 6x 15y + 3z = 6 8x + 20y 4z = 8 12x + 30y 6z = 12 3 x + y + 5 z = 3 3x 5y + 4z = 3 9x 15y 17z = a multiple of ( -3, 1, 0, 0, 0, 0 ) a multiple of ( -6, 0, 4, 1, 0, 0 ) a multiple of ( 11, 0, -4, 0, 1, 1 ) 82. (1 pt) Library/TCNJ/TCNJ SolutionSetsLinearSystems- /problem1.pg Find a set of vectors u, v in R 4 that spans the solution set of the equations: u = x y w = 0 3x + 2y + z + 3w = 0, v =. \(\displaystyle\left.\beginarray}c} \mbox-1} \cr \mbox-6} \cr \mbox0} \cr \mbox5} \cr,\(\displaystyle\left.\beginarray}c} \mbox-1} \cr \mbox-1} \cr \mbox5} \cr \mbox0} \cr 83. (1 pt) Library/TCNJ/TCNJ SolutionSetsLinearSystems- /problem8.pg Suppose the solution set of a certain system of equations can be described as x 1 = 1 + 5t, x 2 = 2 + 4t, x 3 = 6 + 4t, where t is a free variable. Use vectors to describe this set as a line in R 4.

15 +t (1 pt) Library/TCNJ/TCNJ SolutionSetsLinearSystems- /problem9.pg Given A = find one nontrivial solution of A 0 by inspection (1 pt) Library/TCNJ/TCNJ SolutionSetsLinearSystems- /problem6.pg Let A = Describe all solutions of A 0. x 2 +x 3 +x 4 a multiple of ( -3, 1, 0, 0 ) a multiple of ( -4, 0, 1, 0 ) a multiple of ( 1, 0, 0, 1 ) 86. (1 pt) Library/TCNJ/TCNJ RowReduction/problem4.pg 88. (1 pt) Library/TCNJ/TCNJ RowReduction/problem8.pg For the following system to be consistent, 6x + 5y 3z = 5 7x 8y + k z = 3 34x + 37y 23z = 18 we must have, k (1 pt) Library/TCNJ/TCNJ RowReduction/problem3.pg Determine all values of h and k for which the system 5x + 5y 7z = 3 4x + 7y 7z = 2 31x + 13y + hz = k has no solution. k h = (1 pt) Library/TCNJ/TCNJ RowReduction/problem11.pg Let k,h be unknown constants and consider the linear system: 4x + 5y 5z = 4 5x + 8y + 2z = 6 6x 21y + hz = k This system has a unique solution whenever h. If h is the (correct) value entered above, then the above system will be consistent for how many value(s) of k? A. no values B. a unique value C. infinitely many values 6x x 35 k For the above system of equations to be consistent, k must equal (1 pt) Library/TCNJ/TCNJ RowReduction/problem7.pg If the following system 4x x + k 22 is consistent, then k B 91. (1 pt) Library/TCNJ/TCNJ RowReduction/problem12.pg Let k,h be unknown constants and consider the linear system: x + 3 h 8x + k 10 This system has a unique solution whenever k. If k is the (correct) value entered above, then the above system will be consistent for how many value(s) of h? A. no values B. infinitely many values

16 C. a unique value 4 C 92. (1 pt) Library/TCNJ/TCNJ RowReduction/problem5.pg Suppose that the following 12x x 14 k 20 x is a consistent system. Then k = (1 pt) Library/TCNJ/TCNJ RowReduction/problem9.pg If the following system is consistent, 4x k x then k (1 pt) Library/TCNJ/TCNJ RowReduction/problem10.pg If the following system has infinitely many solutions, 6x + 5y 7z = 7 5x 7y + 2z = 9 8x + 29y + hz = k then k =, h = (1 pt) Library/TCNJ/TCNJ RowReduction/problem6.pg If there are an infinite number of solution to the system 7x + 9 h 8x + k 1 then k =, h = (1 pt) Library/TCNJ/TCNJ MatrixEquations/problem2.pg Perform one step of row reduction, in order to calculate the values for x and y by back substitution. Then calculate the values for x and for y. Also calculate the determinant of the original matrix. You can let webwork do much of the calculation for you if you want (e.g. enter 45-(56/76)(-3) instead of calculating the value out). You can also use the preview feature in order to 16 make sure that you have used the correct syntax in entering the answer. Note since the determinant is unchanged by row reduction it will be easier to calculate the determinant of the row reduced matrix. 1-3 x 4 3 y = -1 x y 0 det = = - 4*-3/ *1/1 (1/1) - (-3/1)* (1*-1-4*1)/(1*3 - -3*4) *3 - -3*4 97. (1 pt) Library/TCNJ/TCNJ MatrixEquations/problem4.pg Let A = and ? 1. What does Ax mean? Linear combination of the columns of A 98. (1 pt) Library/TCNJ/TCNJ MatrixEquations/problem11.pg To see if b = 3 16 is a linear combination of the vectors 6 a 1 = 4 3 and a 2 = one can solve the matrix equation A c where the columns of A are v 1 = and v 2 = 1 and c = \(\displaystyle\left.\beginarray}c} \mbox4} \cr \mbox3} \cr \mbox3} \cr,\(\displaystyle\left.\beginarray}c} \mbox6} \cr \mbox10} \cr \mbox-6} \cr \(\displaystyle\left.\beginarray}c} \mbox3} \cr \mbox16} \cr \mbox6} \cr.

17 99. (1 pt) Library/TCNJ/TCNJ MatrixEquations/problem12.pg 5 To see if b = is a linear combination of the vectors a 1 = and a -1 2 = 7 one can solve the matrix equation A c where the columns of A are v 1 = and v 2 = and c =. \(\displaystyle\left.\beginarray}c} \mbox2} \cr \mbox-1} \cr,\(\displaystyle\left.\beginarray}c} \mbox10} \cr \mbox7} \cr \(\displaystyle\left.\beginarray}c} \mbox5} \cr \mbox-5} \cr 100. (1 pt) Library/TCNJ/TCNJ MatrixEquations/problem10.pg -1 Let A be a 3x2 matrix. Suppose we know that u = and -4 1 v = satisfy the equations Au = a and Av = b. Find a solution x to A 4a + 2b (1 pt) Library/TCNJ/TCNJ Dets CramersRule Misc- /problem2.pg 102. (1 pt) Library/TCNJ/TCNJ Dets CramersRule Misc- /problem1.pg Solve the system using Cramer s Rule. 5x 2 2 6x 1 det = (1 pt) Library/TCNJ/TCNJ MatrixInverse/problem17.pg For each section, find the matrix X solving the given equation a. X = X = b. X = X = c. X = X = d. X = X = e X = f X = X = X = Solve the system using Cramer s Rule. 2x + 6y + 25z = 4 9x 23y 97z = 5 6x + 18y + 78z = 5 det = z = g X = X =

18 (1 pt) Library/TCNJ/TCNJ VectorEquations/problem2.pg Write a vector equation x+ y+ z = that is equivalent to the system of equations: 7x + y 3z = 9 3x + 9y + 7z = 9 2x 3y 4z = Generated by c WeBWorK, Mathematical Association of America 18