Linear Algebra Questions and Answers

2 8 points Determine if T is a linear transformation T R R T x y x y x y

A matrix is give 18 9 01 a Determine whether the matrix is in row echelon form Yes No b Determine whether the matrix is in reduced row echelon form Yes No c Write the system of equations for which the given matrix is the augmented matrix Enter each answer in terms of x ar 9 7

The system of linear equations has a unique solution Find the solution using Gaussian elimination or Gauss Jordan elimination y z 8 x 2x 3y 2z 16 3z 4 x y z 4x y

State the dimension of the matrix 21 18

Problem 2 10 points Let f x x1 5 3x 22x 99 a Find the critical points of f x on 1 2

3 Consider the language L ab2c2dkdli 0 j 0 k 1 Is this language context free If so give a PDA for the language If not prove it is not a CFL

Cosine Law Practice a 2 Find the length of AC to 1 decimal place 3 b 3 60 B C 75 B 4 3 Find the measure of ZA B

A Find the standard matrices A and A for T T2 T1 and T T1 T2 A T R R T x y x 2y 5x 4y T2 R2 R2 T2 x y 0 x B 00 00

Consider the region bounded by the graph of y 3x 1 the x axis and the lines x 5 x 9 A reasonable upper bound for the area of this region is

Gray Wolves Revisited The United States Fish and Wildlife Service regularly monitors trends in the population of endangered species By modeling changes in the population they can determine if intervention is necessary for the survival of certain species Your work in class examined the gray wolf a protected species that currently resides in Minnesota Wisconsin and Michigan 2 Populations in the three states are monitored regularly to determine if the populations are experiencing growth or decline The gray wolf population in Minnesota was 3 020 in 2004 and 2 278in 2016 3 2U S Fish and Wildlife Service 2020 Gray wolves Western Great Lakes states Retrieved from https www fws gov midwest wolf history index html 3U S Fish and Wildlife Service 2020 Gray wolves Western Great Lakes states Retrieved from https www fws gov midwest wolf population mi wi nos html Assume that the gray wolf population in Minnesota can be modeled using a linear function Part A Is the gray wolf population increasing or decreasing Make sure you can explain why decreasing there is not enough information to tell increasing Part B What is the rate of change of the wolf population per year Round your answer to the thousandths place Answer wolves per year Part C Write a formula for the linear model M t that predicts the gray wolf population in Minnesota t years after 2004

A line contains the points 2 1 and 6 5 Another line contains the points 3 2 and 2 7 Are the lines parallel Yes the lines are parallel

et 7 R4 R3 be the linear transformation represented by T x Ax where A 1 210 100 124 001 a Find the dimension of the domain 4 b Find the dimension of the range 2 X c Find the dimension of the kernel 0 X d Is T one to one Explain T is one to one since the ker 7 0 T is one to one since the ker 7 0 T is not one to one since the rank 7 0 T is not one to one since the ker 7 0 T is not one to one since the ker 7 0 e Is T onto Explain T is onto since the rank 7 is equal to the dimension of the domain OT is not onto since the rank 7 is not equal to the dimension of the domain T is not onto since the rank 7 is equal to the dimension of the co domain T is not onto since the rank 7 is not equal to the dimension of the co domain T is onto since the rank 7 is equal to the dimension of the co domain f Is T an isomorphism Explain Select all that apply T is not an isomorphism since it is not onto T is not an isomorphism since it is not one to one OT is an isomorphism since it is one to one and onto X

Let T R2 R2 such that T 1 0 1 0 and T 0 1 0 0 a Determine T x y for x y in R b Give a geometric description of T OT is the projection of every vector in R2 onto the z axis OT maps every vector in R to its orthogonal projection in R OT is the projection of every vector in R2 onto the y axis OT maps every vector in R2 to its orthogonal projection in the xy plane OT is the projection of every vector in R onto the x axis

Define the linear transformation T by T x Ax A 23 a Find the kernel of T If there are an infinite number of solutions use t as your paramete ker 7 b Find the range of T O 3t t t is any real number O R OR O t t t is any real number 1

Determine whether the linear transformation is one to one onto or neither Select all that apply T R4 R4 T x y z w w z y x one to one onto

Stellar Inc manufactures cycling equipment Recently the vice president of operations of the company has requested construction of a new plant to meet the increasing demand for the company s bikes After a careful evaluation of the request the board of directors has decided to raise funds for the new plant by issuing 3 128 500 of 12 term corporate bonds on March 1 2020 due on March 1 2035 with interest payable each March 1 and September 1 with the first interest payment on September 1st 2020 At the time of issuance the market interest rate for similar financial instruments is 8 Click here to view factor tables As the controller of the company determine the selling price of the bonds Round factor values to 5 decimal places e g 1 25124 and final answer to O decimal places e g 458 581 Selling price of the bonds

The voltage v t in a network is defined by the equation below given a 2 b 9 and c 2 dv t adu 1 b du t a dt b du 1 v 1 0 Simplify the given equation to the characteristic equation What is the q in the characteristic equation of the network in the form s qs r

A Find the kernel and the range of the linear transformation 1 5pt T R R T x y R T x y z z y x 1 2 1 2 5pt A 0 2 1

128 d e 68 C Calculate the values of c d a

Using V 9 mV and given the voltage across an inductor L 10 mH as shown in Fig P6 25 determine the waveform for the inductor current during the time t1 t t2 where t1 4 ms and t2 19 ms Assuming the solution looks like i t X t t1 2 Y t t1 C what is the initial condition of the current the constant C For your answer just put the magnitude of the initial value of the constant and in milliAmps there should only be a number in your answer D f MV V t1 t2 ms Figure P6 25

Given a capacitor of C 23 F calculate the current going through the capacitor during the interval t2 t t3 when the capacitor voltage is as described in Fig 6 12 Assume V1 9V V2 7V t2 90 s and t3 185 s Note The values might be different than the above problem v t V V1 V2 t2 t3 0 t1 t s Figure P6 12 Notes on entering solution Enter your solution in Amps Pay attention to the interval Enter your solution to 2 decimal points Do not include units in your answer ex 5A is entered as 5 00

Solve the inequality Enter your answer using interval notation x4 34x272 0

Solve the inequality Enter your answer using interval notation x 2 X 3 X 1 X 2

c 2 d y 2 dx 3 2 dy 2 X y 14xe Solve the dx differential equation

sin 1 60 8 61 and 6 tan 9 40 1 Find the exact value of Cos 28 Express your answer as a fraction in lowest terms

On a test of 73 items Pedro got 90 correct There was partial credit on some items How many items did he get correct incorrect Pedro got 65 70 items correct Type a whole number or decimal rounded to two decimal places as needed Pedro got items incorrect Type a whole number or decimal rounded to two decimal places as needed

Translate to an equation and solve What is 55 of 278 is 55 of 278 Simplify your answer Type an integer or a decimal

Bill got 45 correct answers on a 75 question test What percent of the questions answer incorrectly Bill answered of the questions incorrectly

In 2016 there were about 43 500 cinema screens in a country If about 33 1 of the total screens in the country were digital 3 D screens find the approximate number of digital 3 D screens There were approximately digital 3 D screens Round to the nearest whole number as needed

On a test of 79 items Pedro got 90 correct There was partial credit on some items How many items did he get correct incorrect Pedro got items correct Type a whole number or decimal rounded to two decimal places as needed

Use the functions fand g in C 1 1 to find f g f g and d f g for the inner product f g f x g x dx f x 1 g x 4x 1 a f g b fl c g d d f g

Use the inner product p q aobo ab a2b2 to find p q pl a and d p q for the polynomials in P2 p x 1 x q x 1 x a p q b p c a d d p q W

Find the vector v with the given length and the same direction as u V v 3 u 0 1 1 3 3 0 II 3 11 11 9 11

Determine whether u and v are orthogonal parallel or neither u 2 20 v 5 1 O orthogonal O parallel O neither

Which is the only function that could have a range of all real number i e 0 0 A linear function A quadratic function O An exponential function None of the functions could have a range of all real numbers

0 9968 0 9970 gravity of H 0 Tabulation of density specific gravity of water at various Temp 0 9973 0 9975 0 9978 0 9980 0 9982 0 9984 0 9986 0 9987 0 9989 0 4991 19 20 21 22 23 24 Temp C 125 26

Let A and B are square matrices of order 4 such 1A 2 and 1B 5 Find the following AB b A c 43 d AB e A f A a

Prove det AB det A det B

T Prove AB AT BT

Let A be nxn matrix each of whose rows adds P to zero Find A

Find the determinant of the matrex 0 3 3 0 5 0 3

Find A if 3A 4 3

Prove the following statement If A is invertible matrex then AT A

Determine the mean median mode and midrange of the set of data 1 9 13 27 35 15 11 8 1 What is the mean

A food company is testing 5 chocolate chip cookies 4 crackers and 7 reduced fat cookies If it plans to market 3 of the chocolate chip cookies 2 of the crackers and 1 of the reduced fat cookies how many different combinations are possible There are possible combinations Tyne a whole number

Find the real solutions of the following equation by factoring 3 2 2x 10x x 5

Find the perimeter of the rectangle 3 cm by 15 cm

Find the length of the diameter the circumference and the area of the circle 22 Use for 7 78 in The diameter is Simplify your answer Type a whole number or a simplified fraction The circumference of the circle is approximately Simplify your answer Type a whole number or a simplified fraction The area of the circle is approximately Simplify your answer Type a whole number or a simplified fraction

A lot is 60 m by 35 m A house 28 m by 9 m is built on the lot How much area is left over

3 A loan of 100 000 must be repaid in monthly installments of 599 55 over 30 years The bank compounds interest every month What is the nominal annual rate of interest on this loan