Linear Algebra Questions and Answers

The graph of y f x is shown below Select the correct answer holour IN WA 7 6 5 4 3 2 1 R D 1 2 3 4 5 6 7 Which of the answer choices shows the graph of y f 2x 4 3

Use elementary row operations to reduce the given matrix to row echelon form and reduced row echelon form 4 11 2 5 10 26 2 5 1 a row echelon form b reduced row echelon form 3 00

Determine by Inspection 1 e without performing any calculations whether a linear system with the given augmented matrix has a unique solution infinitely many solutions or no solution 3 4 01 1 1 4 3 1 1 2 8 6 2 0 unique solution O Infinitely many solutions Ono solution Justify your answer O The matrix can be rewritten in row echelon form with one free variable The matrix can be rewritten in row echelon form with two free variables O This system is a homogeneous system with four variables and only three equations so the rank of the matrix is at most 3 and thus there is at least one free variable O The matrix can be rewritten so that you have a row 00 00 a with a 0 OTHE no free variables

Find the solution set of the equation If your answer is dependent use the parameters s and t as necessary If there is no solution enter NO SOLUTION 8x 5x 4x3 7 X X2 X3 7 8t 4u 5 9 7 8t 5s 4 X

Show that the given matrices are row equivalent and find a sequence of elementary row operations that will convert A into B 3 2 A B 2 2 A 22 R R1 3 1 23 10 R H R R B

Determine whether the given matrix is in row echelon form O The matrix is in row echelon form The matrix is not in row echelon form 0 17 0 0 0 0 1 If it is state whether it is also in reduced row echelon form The matrix is in reduced row echelon form O The matrix is in row echelon form but is not in reduced row echelon form O The matrix is not in row echelon form

Which function below has the following domain and range Domain 6 5 1 2 6 Range 2 3 8 6 2 5 3 1 8 2 5 6 9 2 3 5 2 1 8 6 3 6 2 O 2 5 8 1 3 6 2 6 3 2 O 6 6 2 8

e movement of the progress bar may be A student correctly evaluated an expression with p 2 and q 3 and got 3 as the result Which of the following expressions could she have been evaluating p 3q q p Op 9 49 Op 2p q p 2pq q 3p 2pq 69 2

Consider the following linear system d 5 a b c 4d 1 Create the augmented matrix of this system 1 a 2b 0 0 1 2 1 0 Convert the system into one that exhibits the triangular structure where the represent numerical values 1 2 0 1 1 11 1 4 3 1 5 6 5 Write the linear system corresponding to the simplified matrix above then solve the linear system If your answer is dependent use the parameters s and t as neces solution enter NO SOLUTION a b c d

Consider the following linear system d 5 4d 1 a 2b a b c Create the augmented matrix of this system 1 2 1 1 5 Convert the system into one that exhibits the triangular structure 1 01 where the represent numerical values Write the linear system corresponding to the simplified matrix above then solve the linear system If your answer is dependent use the parameters s and t as nece solution enter NO SOLUTION a b c d

Determine if the equation is a linear equation in the variables x y and z If the equation is not linear explain why not sin 1 x 5y z sin O The equation is a linear equation O The equation is not a linear equation since it contains a trigonometric function O The equation is not a linear equation since it contains a product of two variables O The equation is not a linear equation since at least one of x y or z has a power other than 1 The equation is not a linear equation since at least one of x y or z has a coefficient other than 1

Find a system of linear equations that has the given matrix as its augmented matrix O x y z 1 x y z 1 8x y z 1 0 1 1 1 1 1 0 1 8 1 11 y z 1 x y 1 8x y z 1 y 8z 1 z 1 Z 1 Ox y X 8x y 1 O x y z 1 X z 1 x y 8z 1 x y Z 1 x y z 1

Find the augmented matrix of the linear system 2X 3X X3 3 x3 0 X1 X 2x 3x3 0 2 1 3

Points DETAILS area of a sector of a circle with a central angle of 4 rad eed Help Read It Points DETAILS Watch It

In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equation dP dt P a bp where a and b are positive constants Although we will come back to this equation and solve it by an alternative method in Section 3 2 solve the DE this first time using the fact that it is a Be equation P t

Tork University s Learning X UAssignment 2 pag D eclass yorku ca mod quiz attempt MATH1507 M N C

vey has shown that 42 of the adults in Mystic Falls have vision insurance Of these adults 55 have iPhone while 33 of the adults who do not have vision insurance have iPhones Find the probabilities rounded to 3 significant digits that an adult in Mystic Falls can be categorized as follows adult who has an iPhone adult who does not have an iPhone but has vision insurance e the events that an adult who has vision insurance and an adult who has an iPhone independent

0 1 V Exercise 5 8 4 Consider the two vectors U Think of these both as 3 x 1 matrices Which of the following multiplications are defined v v2 v v2 v v or uf u Which one of these is equal to the inner product Note that the remaining one is called the outerproduct In reality we should have defined the inner product from Definition 3 1 2 as v 02 v v2 Keep this in mind for the future

Exercise 5 8 5 Draw 5 dots on a piece of paper and label them po P1 P2 P3 P4 Decide to connect some of these dots and not others Now make a matrix which has the same number of columns as there are dots and the same number of rows as there are connections you drew Label the rows for the connections like 01 03 14 etc Now for each connection you make from p to p in order so that i j for example if you connect p4 and po we will call this po to p4 in the corresponding row put a 1 in the column for p and a 1 in the column for p and 0 in all other columns Now study the kernel of this matrix Its dimension will equal the number of connected components your picture had For example if all the pieces form one cluster you will have one kernel dimension Try to do this a second time so that your picture has two distinct clusters and see if you can reflect that in your computation of the kernel This is a warm up for one of our final projects

elect the correct choice that completes the s a modulo m system the number m is called

Solvere following linear system by the elimination method if possible if does not have a solution leave answers blank 0 3x 0 7y 1 3 2x 0 9y 0 2 The value of x is keep answer as reduced fraction enter or sign for the value of x numerator of x is denominator of x is The value of y is keep answer as reduced fraction enter or sign for the value of y numerator of y is denominator of y is

If you need to use to indicate a pattern make sure to list at least four a Set builder form x x is an integer and 3 x 6 Roster form 0 b Roster form 4 3 2 1 Set builder form Choose one Write each set in the indicated form If you need to use to indicate a pattern make sure to list at least four el a Set builder form x x is an integer and 3 r 6 Roster form yy is an integer and y 5 yy is an integer and y 5 fyly is an integer and 4 v 1

Mary Branson invested her summer earnings of 5000 in a savings account for college The account pays 6 5 interest How much will this amount to in 3 months in 3 months Mary s savings will amount to Round to the nearest cent as needed

vector function is a solution to a matrix differential equation if after substituting into the matrix differential equation O A it makes both sides of the matrix DE not equal B it makes another matrix DE OC it solves the equation for x and y OD it makes both sides of the matrix DE equal

pan Show that R2 span We must show that for any vector 8 4 8 for some x y Row reduce the associated augmented matrix a BOBAN 18 R R 1 b R R So given a and b we have Iwe can write 10 0 1 1 1 0 2 DELC R 1 1 01 1 DETAL

Consider the following 00 HH S Hi 00 0 0 52 A A11B12 A12B22 AB A11 A12 A21 A22 1 Compute each of the following using the indicated partitioning A11B11 A12B21 B11 B12 B21 B22 A21B11 A22B21 A21B12 A22B22 Compute AB by block multiplication using the indicated partitioning 52 1 1 00 00 0 1

Write the equation of the line passing through P with normal vector n in normal form and general form P 2 1 n a normal form b general form

Give bases for row A col A and null A 1 1 6 0 2 1 1 1 7 row A col A null A A 11

Let TA 1 3 u 2 and v 2 TA u TA V R2 R be the matrix transformation corresponding to A 4 1 2 3 Find TA u and T v where

y x T is a linear transformation if and only if Then entering your answers PM T C V C V C T v C T v where v in terms of 4 X3 2 13 ebassign net v4cgi assignments preview tpl aid 21237117 UserPass 966756bba9e613844df029aba63e370f C C X X2 X1 and Y2 T C V C V TC x 2 Assignment Previewer C2X2 C22

Consider the following set of vectors 0 0 V 1 9 C1 Let v and v be column vectors in R3 and let A be the 3 x 2 matrix V with these vectors as its columns Then v and v are linearly dependent if and only if the homogeneous linear system with augmented matrix A10 has a nontrivial solution Consider the following equation 9 C 1 7 9 0 0 0 7 9 Solve for C and C If a nontrivial solution exists state it or state the general solution in terms of the parameter t If only the trivial solution exists enter the trivial solution C C 0 0 C C Determine if the vectors v and v are linearly independent O The set of vectors is linearly dependent O The set of vectors is linearly independent

What is the rank of each of the matrices below a 44 PM b c d e f 1 0 1 006 0 1 0 9 90 10 0 1 1 7 00 00 0160 0 0 0 1 0 0 0 0 0 0 0 1 0 6 7 0 000 00 0 1 8 0 1 w webassign net v4cgi assignments preview tpl aid 21237117 UserPass 966756bba9e613844df029ab 0 0 1 0 1 0 1 0 0 1 2 6 1 0 0 0 1 1 0 h 2 16 8 Assignm

Let 30 0 3 A 1 D 6 Compute the indicated matrix If this is not possible enter DNE in any single blank A 3D

3 1 1 Write next to the image whether you would use Heartbeat H or Boomerang B 2 Set up the equation to solve for x Do Not Solve 48 64 25 2 4 7 81 45 9

11 Perhatikan gambar 14 cm Diketahui volume tabung 2 156 cm Luas alas tabung tersebut adalah 64 cm 89 cm a b c d 154 cm 308 cm

If you have y e is a solution of xy x 10 y 10y 0 fi the second solution to this differential equation

In Exercises 13 16 use a rectangular coordinate system to plot u 1 2 v 2 and their images under the given transformation T Make a separate and reasonably large sketch for each exercise Describe geometrically what I does to each vector x in R 15 T x 1 0 01 X1 16 T x

Find a 2 x 2 matrix A for which 3 where Ex denotes the eigenspace associated with the eigenvalue A 1001 E 5 span E3 2 span

If the profit from the sale of x units of a product is P 105x 300 x2 what level s of production will yield a profit of 2150 Enter your answers as a comma separated units

Solve the equation by factoring Enter your answers as a comma separated list 25 36x X 0

The monthly profit from the sale of a product is given by P 14x 0 1x 150 dollars a What level of production maximizes profit units b What is the maximum possible profit

5 For this question consider the linear transformation T P4 M2x2 given by T ax bx cx dx e 2a2b c e a b 2c 3d e a b 2c e e c d e You may assume T is linear a What is the domain and codomain of T b Let B x x x x 1 and 1 8 8 5 i 9 the matrix T B c Choose u x x x Verify the formula T of 5 b Find TB Bus using the bases

2 5pts There is an urn containing 10 marbles numbered 1 to 10 We sample 4 marbles without replacement a Express the sample space b Let E be an event that the 4 marbles contain numbers 1 and 2 Express the event E and calculate how many elements E contains

Suppose that the point x y is in the indicated quadrant Decide whether the given ratio is positive or negative Recall that r x 11 2 In Quadrant II is positive or negative O Negative O Positive

Part A Graph the following constraints These represent the constraints on my business for one day Constraints y 2 x 7 y 2x 10 x 0 y 20 Part B List the vertices of the feasible region 10 9 8 7 6 5 4 3 2 1 y 0 1 2 3 4 5 6 7 8 9 10 Part C The objective function P 4x y represents my profit for the day in thousands of ollars What is the maximum and minimum profit for the day and explain what it means

Congratulations you have worked out how row reduction is also a matrix factoring algorithm Exercise 3 6 7 Using the ideas from the above exercise go back to exercise 1 4 8 and find the matrix E which is a product over a bunch of matrices coming from elementary row operations so that E A R where A is the coefficient matrix for the problem and R is the final reduced matrix you used

1 2 Solve for all matrices C so that BC C B The proof of theorem 3 2 5 depends on how one approaches the subject it can either be a definition or a theorem But for us one would have to write out an arbitrary matrix and check the properties From there we can deduce an additional property as explained below Exercise 3 6 2 Consider the matrix B Theorem 3 6 3 If A E Matmxn and 0 Proof O A 0 ER then A 0 0 Rm Othm 325 a

30h h 16 1 Suppose that R h represents the number of people in hundreds waiting in line for a ride at a theme park h hours after the park opens Evaluate and interpret R 8 and R 8 using complete sentences

Two companies have offered you a job Alpha Company has offered you 31 000 per year with an annual raise of 2800 Beta Company has offered you a 39 000 annual salary with an annual raise of 1100 per year Which company will pay you more over the first ten years Alpha Company will pay 56 200 over a ten year period Beta Company will pay 48 900 over a ten year period

The ratio of the perimeter of a regular hexagon to the length of one of its sides is always 6 to 1 Complete the table Perimeter of hexagon inches 6 30 48 Side length of hexagon inches Graph the data from the table 10 1 9