This file is /conf/snippets/setheader.pg you can use it as a model for creating files which introduce each problem set.

Transcription

1 Ivan Ivanov WeBWorK assignment number hw is due : 9/7/ at :am EDT The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information linalg This file is /conf/snippets/setheaderpg you can use it as a model for creating files which introduce each problem set The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help Don t spend a lot of time guessing it s not very efficient or effective Give or significant digits for (floating point) numerical answers For most problems when entering numerical answers, you can if you wish enter elementary expressions such as instead of 8, sin( pi/)instead of -, e (ln()) instead of, ( +tan()) ( sin()) 6 7/8 instead of 76, etc Here s the list of the functions which WeBWorK understands You can use the Feedback button on each problem page to send to the professors ( pt) rochesterlibrary/setlinearalgebrasystems/ur la bpg Solve the system using matrices (row operations) { x+y 8 x y 9x 8y 8 ( pt) rochesterlibrary/setlinearalgebrasystems/ur la apg Solve the system using matrices (row operations) x y z x+y+z x+y z x y+6z ( pt) rochesterlibrary/setlinearalgebrasystems/ur la 6pg For each system, determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions { x+y 6 x 8y A No solutions B Infinitely many solutions C Unique solution: x 6, y D Unique solution: x, y E Unique solution: x, y 6 F None of the above { x +y x y 6 A No solutions B Infinitely many solutions C Unique solution: x, y D Unique solution: x, y 6 E Unique solution: x, y F None of the above { 9x+y 7 8x+9y A No solutions B Unique solution: x, y 7 C Infinitely many solutions D Unique solution: x, y E Unique solution: x 7, y F None of the above { 8x+6y 7x+y A Infinitely many solutions B Unique solution: x, y 8 C Unique solution: x, y D No solutions E Unique solution: x, y F None of the above ( pt) rochesterlibrary/setlinearalgebrasystems/ur la 8pg Determine the value of h such that the matrix is the augmented matrix of a consistent linear system h 8 6 ( pt) rochesterlibrary/setlinearalgebrasystems/ur la 9pg Determine the value of h such that the matrix is the augmented matrix of a linear system with infinitely many solutions 8 6 h h 8

2 h 6 ( pt) rochesterlibrary/setlinearalgebrasystems/ur la pg The reduced row-echelon forms of the augmented matrices of four systems are given below How many solutions does each system have? -9 A No solutions B Infinitely many solutions C Unique solution D None of the above -7 A Unique solution B No solutions C Infinitely many solutions D None of the above A Unique solution B Infinitely many solutions C No solutions D None of the above - A Infinitely many solutions B Unique solution C No solutions D None of the above 7 ( pt) rochesterlibrary/setlinearalgebrasystems/ur la 6pg Solve the system x y x +y 7 x y x 9y 8 ( pt) rochesterlibrary/setlinearalgebramatrices/ur la pg - -9 Reduce the matrix - - to reduced row-echelon form 9 ( pt) rochesterlibrary/setlinearalgebramatrices/ur la pg Reduce the matrix to reduced row-echelon -8 form ( pt) rochesterlibrary/setlinearalgebrasystems/ur la pg Determine the value of k for which the system has no solutions k x +y+z x +y z 6x+y+kz ( pt) rochesterlibrary/setlinearalgebrasystems/ur la pg Solve the system { x +x x x +x x 9 x x x + s ( pt) rochesterlibrary/setlinearalgebrasystems/ur la 8pg Solve the system x x x x x +x x +x x +x x +x + s ( pt) rochesterlibrary/setlinearalgebrasystems/ur la 9pg Solve the system x x x x x x +x +x x +x +x +x x x +x +x x +x +6x +6x 9 + s + t

3 ( pt) rochesterlibrary/setlinearalgebrasystems/ur la pg Solve the system x x x x x x 6 + u x +x +x +x x 6 x x x 6 9 x +x 6x +6x s + ( pt) rochesterlibrary/setlinearalgebrasystemsapplications- /ur la pg Find the quadratic polynomial whose graph goes through the points (,), (,), and (,6) f (x) x + x+ 6 ( pt) rochesterlibrary/setlinearalgebrasystemsapplications- /ur la pg Find the cubic polynomial f (x) such that f (), f (), f (), and f () 6 f (x) x + x + x+ 7 ( pt) rochesterlibrary/setlinearalgebrasystemsapplications- /ur la pg Consider the chemical reaction aal + bh O cal(oh) + dh, t 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, b c 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 8 ( pt) rochesterlibrary/setlinearalgebramatrices/ur Ch pg Compute the rank of the above matrix 9 ( pt) rochesterlibrary/setlinearalgebramatrices/ur la pg Findthe ranks of the following matrices rank - - rank rank - ( pt) rochesterlibrary/setlinearalgebramatrices/ur la 6pg Find the value of k for which the matrix A 9 9 has rank 9 k k Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

4 Ivan Ivanov WeBWorK assignment number hw is due : 9// at :am EDT The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information linalg This file is /conf/snippets/setheaderpg you can use it as a model for creating files which introduce each problem set The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help Don t spend a lot of time guessing it s not very efficient or effective Give or significant digits for (floating point) numerical answers For most problems when entering numerical answers, you can if you wish enter elementary expressions such as instead of 8, sin( pi/)instead of -, e (ln()) instead of, ( +tan()) ( sin()) 6 7/8 instead of 76, etc Here s the list of the functions which WeBWorK understands You can use the Feedback button on each problem page to send to the professors ( pt) Library/Rochester/setLinearAlgebraMatrices/ur la pg If A - - and B Then A + B and A T ( pt) Library/TCNJ/TCNJ MatrixOperations/problempg Solve for X X -9-6 X Then AB and BA 6 ( pt) Library/Rochester/setLinearAlgebraMatrices/ur la 8pg Compute the following products ( pt) Library/Rochester/setLinearAlgebraMatrices/ur la 6pg If A and B are 7 6 matrices, and C is a 8 7 matrix, which of the following are defined? ( pt) Library/Rochester/setLinearAlgebraMatrices/ur la 7pg Compute the following product ( pt) Library/Rochester/setLinearAlgebraMatrices/ur la 8pg Compute the following product - ( pt) Library/Rochester/setLinearAlgebraMatrices/ur la pg If A - - and B A BA B C T C B T C T D C + A E B A F CA 8 ( pt) Library/Rochester/setAlgebraMatrices/sw7 7pg - Given the matrix A, find A - Write A a a as a a Input your answer below: a a a a

5 9 ( pt) Library/Rochester/setLinearAlgebraInverseMatrix- /ur la pg -7-8 If A - -9 Then A ( pt) Library/Rochester/setLinearAlgebraInverseMatrix- /ur la pg - The matrix is invertible if and only if k 7 k ( pt) Library/Rochester/setLinearAlgebraInverseMatrix- /ur la pg - - If A Then A ( pt) Library/Rochester/setLinearAlgebraInverseMatrix- /ur la 6pg If A 8 - Then A ( pt) Library/Rochester/setLinearAlgebraInverseMatrix- /ur la 8pg Determine which of the formulas hold for all invertible n n matrices A and B A (ABA ) 6 AB 6 A B ABA B C A B 8 is invertible D (A + A ) A + A E A + B is invertible F (A + B) A + B + AB ( pt) Library/Rochester/setLinearAlgebraInverseMatrix- /ur la pg e If A t sin(9t) e t cos(9t) e t cos(9t) e t sin(9t) then A ( pt) Library/TCNJ/TCNJ MatrixInverse/problempg If A Then A Given b, solve Ax b x 6 ( pt) Library/ASU-topics/set9MatrixAlgebra/ppg Consider the following two systems (a) { x y x + 7y (b) { x y x + 7y (i) Find the inverse of the (common) coefficient matrix of the two systems A (ii) Find the solutions to the two systems by using the inverse, ie by evaluating A B where B represents the right hand side (ie B for system (a) and B for system (b)) Solution to system (a): x, y Solution to system (b): x, y 7 ( pt) Library/TCNJ/TCNJ MatrixOperations/problem6pg Solve for X X X 8 ( pt) Library/TCNJ/TCNJ MatrixOperations/problem8pg Solve for X X+ X X

6 9 ( pt) Library/TCNJ/TCNJ MatrixInverse/problempg Suppose that: A and B Given the following descriptions, determine the following elementary matrices and their inverses a The elementary matrix E multiplies the first row of A by / E, E b The elementary matrix E multiplies the second row of A by - E, E c The elementary matrix E switches the first and second rows of A E, E d The elementary matrix E adds 6 times the first row of A to the second row of A E, E e The elementary matrix E multiplies the second row of B by / E, E f The elementary matrix E 6 multiplies the third row of B by - E 6, E 6 g The elementary matrix E 7 switches the first and third rows of A E 7, E 7 h The elementary matrix E 8 adds times the third row of A to the second row of A E 8, E 8 ( pt) Library/TCNJ/TCNJ MatrixInverse/problempg - -6 a Suppose that: E Find E and E E, E - b Suppose that: E Find E - and E E E, E - c Suppose that: E E, E d Suppose that: E Find E and E E e Suppose that: E Find E and E E , E f Suppose that: E 6 Find E 6 and E6 E , E , E 6 Find E and Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

7 Ivan Ivanov WeBWorK assignment number hw is due : /8/ at :pm EDT The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information linalg This file is /conf/snippets/setheaderpg you can use it as a model for creating files which introduce each problem set The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help Don t spend a lot of time guessing it s not very efficient or effective Give or significant digits for (floating point) numerical answers For most problems when entering numerical answers, you can if you wish enter elementary expressions such as instead of 8, sin( pi/)instead of -, e (ln()) instead of, ( +tan()) ( sin()) 6 7/8 instead of 76, etc Here s the list of the functions which WeBWorK understands You can use the Feedback button on each problem page to send to the professors ( pt) markov/ita9-pg Physical traits are determined by the genes that an offspring receives from its parents In the simplest case a trait in the offspring is determined by one pair of genes, one member of the pair inherited from the male parent and the other from the female parent Typically, each pair in a gene can assume one of two froms, called alleles, denoted by A and a This leads to three possible pairings AA, Aa, aa called genotypes It is shown in a study of heredity that if a parent of known genotype is crossed with a random parent of unknown genotype, than the offspring will have the genotype probabilities given in the following table: AA Aa aa AA Aa aa Here the columns are labelled by the genotypes of the parent and the rows by the genotypes of the offspring Find the steady state vector and think of its physical interpretation The steady state vector is: ( pt) markov/ita9-pg ACME trucking rents trucks in New York, Boston and Chicago The trucks are returned to those cities in accordance with the following table New York Boston Chicago New York 6 Boston 9 Chicago 9 Here the columns are labelled by the locations where the trucks are rented and the rows by the locations where the trucks are returned Determine the distributions of the trucks over the long run: ( pt) rochesterlibrary/setlinearalgebra6determinants- /ur la 6 7pg Determine all minors and cofactors of A M, C, M, C, M, C, M, C, M, C, M, C, M, C, M, C, M, C ( pt) rochesterlibrary/setlinearalgebra6determinants- /ur la 6 9pg If A and B are matrices, det(a), det(b), then det(ab), det( A), det(a T ), det(b ), det(b ) ( pt) rochesterlibrary/setlinearalgebra6determinants- /ur la 6 pg Findthe determinant of the matrix M -7 - det(m)

8 6 ( pt) rochesterlibrary/setlinearalgebra6determinants- /ur la 6 pg Findthe determinant of the matrix M det(m) 7 ( pt) rochesterlibrary/setlinearalgebra6determinants- /ur la 6 pg Findthe determinant of the matrix M - - det(m) 8 ( pt) rochesterlibrary/setlinearalgebra6determinants- /ur la 6 pg If det a d b e and det a d b e, c f c f then det a 6 d b 6 e, c 6 f and det a d b e c f 9 ( pt) rochesterlibrary/setlinearalgebra6determinants- /ur la 6 pg Let A Find the following: (a) det(a), (b) the matrix of cofactors C (c) adj(a) (d) A ( pt) rochesterlibrary/setlinearalgebra6determinants- /ur la 6 6pg e t e Let A t 6e t e t Find the following: (a) det(a), (b) the matrix of cofactors C,,, (c) adj(a), (d) A ( pt) rochesterlibrary/setlinearalgebraeigenvalues- /ur la pg Find the characteristic polynomial of the matrix A - - p(x) ( pt) rochesterlibrary/setlinearalgebraeigenvalues- /ur la pg -9-9 The matrix C has two distinct eigenvalues, λ < λ : λ has multiplicity, and λ has multiplicity ( pt) rochesterlibrary/setlinearalgebraeigenvalues- /ur la 8pg Find the eigenvalues of the following matrix Note you may want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues You can use the web version at xfunctions Also, You can use this rowreduction tool to help with the calculations A The eigenvalues are λ < λ < λ, where λ, associated eigenvector (,, ), λ, associated eigenvector (,, ), λ, associated eigenvector (,, ) ( pt) rochesterlibrary/setlinearalgebraeigenvalues- /ur la 9pg Supppose A is an invertible n n matrix and v is an eigenvector of A with associated eigenvalue Convince yourself that v is an eigenvalue of the following matrices, and find the associated eigenvalues: A 6, eigenvalue, A, eigenvalue, A 7I n, eigenvalue, 7A, eigenvalue ( pt) rochesterlibrary/setlinearalgebraeigenvalues- /ur la pg Find a matrix A such that - and - -

9 are eigenvectors of A, with eigenvalues 6 and 7 respectively A 6 ( pt) rochesterlibrary/setlinearalgebraeigenvalues- /ur la pg Suppose a matrix A has only two distinct eigenvalues Suppose that tr(a) and det(a) Find the eigenvalues of A with their algebraic multiplicities smaller eigenvalue has mulitiplicity, and larger eigenvalue has mulitiplicity 7 ( pt) rochesterlibrary/setlinearalgebraeigenvalues- /ur la pg The matrix A has two real eigenvalues λ < λ Find these eigenvalues, their multiplicities, and dimensions of the corresponding eigenspaces λ has multiplicity The dimension of the corresponding eigenspace is λ has multiplicity The dimension of the corresponding eigenspace is 8 ( pt) rochesterlibrary/setlinearalgebraeigenvalues- /ur la pg Give an example of a x matrix without any real eigenvalues: 9 ( pt) rochesterlibrary/setlinearalgebradiagonalization- /ur la pg Let M -6 Find formulas for the entries of M n, where n is a positive integer M n ( pt) rochesterlibrary/setlinearalgebradiagonalization- /ur la pg 6 Let: A -8-8 Find an invertible S and a diagonal D such that S AS D S D Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

10 Ivan Ivanov WeBWorK assignment number hw is due : // at :am EDT The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information linalg This file is /conf/snippets/setheaderpg you can use it as a model for creating files which introduce each problem set The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help Don t spend a lot of time guessing it s not very efficient or effective Give or significant digits for (floating point) numerical answers For most problems when entering numerical answers, you can if you wish enter elementary expressions such as instead of 8, sin( pi/)instead of -, e (ln()) instead of, ( +tan()) ( sin()) 6 7/8 instead of 76, etc Here s the list of the functions which WeBWorK understands You can use the Feedback button on each problem page to send to the professors ( pt) dynsys/ita9-pg Consider the evolution of the population of a species of lizards Assume that each lizard remains a juvenile for one year and then becomes an adult, and that only adults have offspring The following information is available about the reproduction and survival rates of these lizards: The number of juveniles hatched in any year is on average times the number of adults alive the year before The proportion of adults in any year that survive to the next year is 8 The proportion of juveniles in any year that survive into adulthood is 6 What is the dominant eigenvalue of the matrix associated with this dynamical system? λ dom If initially the population of lizards consisted of adults and juveniles, what is your long term projection for the number of adults and juveniles in this population? What is the dominant eigenvalue of the matrix associated with this dynamical system? λ dom What are the projected long range population ratios between the three age classes? : : ( pt) rochesterlibrary/setvectorsdotproduct/ur VC 8pg Let a (9, -, -) and b (-,, 7) be vectors Compute the following vectors A a + b (,, ) B a (,, ) C a - b (,, ) D a ( pt) rochesterlibrary/setvectorsdotproduct/ur VC pg Find a b if a 6, b 7, and the angle between a and b is π 7 radians a b ( pt) rochesterlibrary/setvectorsdotproduct/ur VC pg ( pt) dynsys/ita9-pg A population of rabbits raised in a research laboratory has the following characteristics Half the rabbits survive their first year Of those, half survive their second year The maximum life span is years During the first year the rabbits produce no offspring The average number of offspring is 6 during the second year and 8 during the third year The laboratory population now consists of 6 rabbits in the first age class, rabbits in the second and in the third How many rabbits will be in each age class after year? If a (6, -, -) and b (, -8, 6), find a b 6 ( pt) rochesterlibrary/setvectorsdotproduct/ur VC pg What is the angle in radians between the vectors a (6,, -) and b (7,, -)? Angle: (radians) 7 ( pt) rochesterlibrary/setvectorsdotproduct/ur VC 6pg A constant force F 6i 6j k moves an object along a straight line from point (-, -, -) to point (-, -9, 9) Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons

11 Work: Nm 8 ( pt) rochesterlibrary/setvectorsplaneslines/ur vc 9apg Find a vector equation for the line through the point P (,, ) and parallel to the vector v (-, -, -) Assume r() i + j + k and that v is the velocity vector of the line r(t) i + j + k Rewrite this in terms of the parametric equations for the line x y z 9 ( pt) rochesterlibrary/setvectorsplaneslines/ur vc pg Consider the planes x + y + z and x + z (A) Find the unique point P on the y-axis which is on both planes (,, ) (B) Find a unit vector u with positive first coordinate that is parallel to both planes i + j + k (C) Use parts (A) and (B) to find a vector equation for the line of intersection of the two planes,r(t) i + j + k ( pt) rochesterlibrary/setvectorsplaneslines/ur vc pg (A) Find the parametric equations for the line through the point P (,, ) that is perpendicular to the plane x y+z Use t as your variable, t should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane x y z (B) At what point Q does this line intersect the yz-plane? Q (,, ) ( pt) rochesterlibrary/setvectorsplaneslines/ur vc pg Consider the two lines L : x t,y + t,z t and L : x + s,y + s,z + s Find the point of intersection of the two lines P (,, ) ( pt) rochesterlibrary/setvectorsplaneslines/ur vc 6pg Find an equation of the plane through the point (-,, ) and parallel to the plane x y + z Do this problem in the standard way or WebWork may not recognize a correct answer x + y + z ( pt) rochesterlibrary/setvectorsplaneslines/ur vc 7pg Find the point P where the line x + t, y t, z -t intersects the plane x + y - z P (,, ) ( pt) rochesterlibrary/setvectorsplaneslines/ur vc 8pg Find the angle in radians between the planes x + z and y + z ( pt) rochesterlibrary/setvectorsplaneslines/ur vc 9pg Find the distance from the point (,, ) to the line x,y + t,z + t 6 ( pt) rochesterlibrary/setvectorsplaneslines/ur vc pg Find the distance from the point (,, ) to the plane x + y + z 7 ( pt) rochesterlibrary/setvectorsdotproduct/ur VC pg Let a (,, -6) and b (, -, 9) be vectors Find the scalar, vector, and orthogonal projections of b onto a Scalar Projection: Vector Projection: (,, ) Orthogonal Projection: (,, ) 8 ( pt) rochesterlibrary/setvectorsdotproduct/ur VC 8pg Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (, ) and arrived in the Iron Hills at the point with coordinates (, 6) If he began walking in the direction of the vector v i + j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn (, ) 9 ( pt) rochesterlibrary/setvectorscrossproduct/ur vc pg Let a (9, 7, 8) and b (7,, ) be vectors Compute the cross product a b (,, ) ( pt) rochesterlibrary/setvectorscrossproduct/ur vc pg If a i + 7j + k and b i + 7j + k, find a unit vector with positive first coordinate orthogonal to both a and b i + j + k ( pt) rochesterlibrary/setvectorscrossproduct/ur vc pg Find the area of the parallelogram with vertices (,), (, ), (, 8), and (6, ) ( pt) unionlibrary/setmvvectors/an pg Suppose u,, and v,, Then: () The projection of u along v is () The projection of u orthogonal to v is

12 ( pt) unionlibrary/setmvvectors/an bpg Find two vectors v and v whose sum is,,, where v is parallel to,, while v is perpendicular to,, v and v ( pt) unionlibrary/setmvvectors/an /an bpg The distance d of a point P to the line through points A and B is the length of the component of AP that is orthogonal to AB, as indicated in the diagram 6 ( pt) unionlibrary/setmvlinesplanes/an 7apg Give a vector parametric equation for the line through the point (, ) that is perpendicular to the line t,t : L(t) 7 ( pt) unionlibrary/setmvlinesplanes/an 6 pg An implicit equation for the plane passing through the points (,, ), (,, 6), and ( 8,, ) is 8 ( pt) unionlibrary/setmvlinesplanes/an 6 7pg The line L(t) 7 t, t,t 8 intersects the plane x+ y + z 7 at the point when t 9 ( pt) unionlibrary/setmvlinesplanes/an 6 7pg The plane that passes through the point (,,) and is perpendicular to both x y + z and y x + z has as its implicit equation (Show the student hint after attempts: ) So the distance from P (,,) to the line through the points A (,, ) and B (,, ) is ( pt) unionlibrary/setmvvectors/an /an cpg Consider the line L(t) t,t, ( + t) and the point P (,,) How far is P from the line L? Hint: How do the normal vectors for two perpendicular planes relate to each other? ( pt) unionlibrary/setmvlinesplanes/planes-pg The planes x y+z and y x z 7 are not parallel, so they must intersect along a line that is common to both of them The vector parametric equation for this line is L(t) (Show the student hint after attempts: ) Hint: How does this relate to the previous problems where you calculated the distance from a point to a line? (Show the student hint after attempts: ) Hint: How is the direction of the line related to the normal vectors for the plane? To find a point on the line, you should find a point that is on both planes Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

13 Ivan Ivanov WeBWorK assignment number hw is due : // at :am EST The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information linalg This file is /conf/snippets/setheaderpg you can use it as a model for creating files which introduce each problem set The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help Don t spend a lot of time guessing it s not very efficient or effective Give or significant digits for (floating point) numerical answers For most problems when entering numerical answers, you can if you wish enter elementary expressions such as instead of 8, sin( pi/)instead of -, e (ln()) instead of, ( +tan()) ( sin()) 6 7/8 instead of 76, etc Here s the list of the functions which WeBWorK understands You can use the Feedback button on each problem page to send to the professors - ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg If T : R R is a linear transformation such that 8 and T T, - - then the standard matrix of T is A ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg If T : R R is a linear transformation such that - and T - T, 9-7 then the standard matrix of T is A ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg Find the inverse of the linear transformation 6 A Reflection in the origin B Rotation through an angle of 9 in the counterclockwise direction C Projection onto the y-axis D Rotation through an angle of 9 in the clockwise direction E Reflection in the line y x F Identity transformation ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg Find the matrix A of the linear transformation T from R to R that rotates any vector through an angle of in the clockwise direction A x y + y, x y + y y x x y x +6x ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg Match each linear transformation with its matrix ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg Find the matrix A of the orthogonal projection onto the line L in R that consists of all scalar multiples of the vector A 6 7 ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg Find the matrix A of the reflection in the line L in R that consists of all scalar multiples of the vector A 6

14 8 ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg 9 8 Let A and B 6 7 Find the matrix C of the linear transformation T (x) B(A(x)) C 9 ( pt) rochesterlibrary/setlinearalgebratransfofrn- /ur la pg - Let T : R R be given by T (x) x Find the matrix M of the inverse linear transformation, T M ( pt) rochesterlibrary/setlinearalgebra6determinantoftransf- /ur la 6 pg Consider a linear transformation T (x) Ax from R to R Suppose for two vectors v and v in R we have T (v ) v and T (v ) v Find the determinant of the matrix A det(a) ( pt) rochesterlibrary/setlinearalgebra8vectorspaces- /ur la 8 pg Which of the following subsets of P are subspaces of P? A {p(t) p(7) } B {p(t) p (t) + 9p(t) + } C {p(t) p (t) is constant } D {p(t) p( t) p(t) for all t} E {p(t) p () p()} F {p(t) p(6) } ( pt) rochesterlibrary/setlinearalgebra8vectorspaces- /ur la 8 pg Which of the following subsets of R are subspaces of R? A The matrices whose entries are all integers B The invertible matrices C The matrices with all zeros in the first row D The matrices whose entries are all greater than or equal to E The matrices A such that the vector is in the kernel of A F The matrices with trace (the trace of a matrix is the sum of its diagonal entries) ( pt) rochesterlibrary/setlinearalgebra8vectorspaces- /ur la 8 6pg Which of the following sets are subspaces of R? A {(x,7x, x) x arbitrary number } B {(x,y,z) x + y + z } C {(x,y,z) 8x + 6y, x + z } D {(x,y,z) x,y,z > } E {(x,y,z) x + y + z } F {(x,x 8,x 6) x arbitrary number } ( pt) rochesterlibrary/setlinearalgebra8vectorspaces- /ur la 8 pg -6 - Let x and y -6 Find the vector v x y and its additive inverse v, v, ( pt) rochesterlibrary/setlinearalgebra9dependence- /ur la 9 8pg Let v -, v -6, and y h For what value of h is y in the plane spanned by v and v? h 6 ( pt) rochesterlibrary/setlinearalgebra9dependence- /ur la 9 pg Express the vector v as a linear combination of - - x and y - - v x+ y 7 ( pt) tcnjlibrary/tcnj VectorSpaces/problempg Let W be the set of all vectors of the form: vectors u and v suchthat W Span{u,v} u, v -s-t s-t s+t s-t Find 8 (pt) tcnjlibrary/tcnj VectorSpaces/problempg Let v, v 9, v and w Is w in {v,v,v }? Type yes or no How many vectors are in {v,v,v }? Enter inf if the answer is infinitely many How many vectors are in Span{v,v,v }? Enter inf if the answer is infinitely many Is w in the subspace spanned by {v,v,v }? Type yes or no

15 9 ( pt) Library/TCNJ/TCNJ VectorEquations/problempg Let H span{v,v,v,v } For each of the following sets of vectors determine whether H is a line, plane, or R? v -, v, v -, v ? v - -, v, v, v ? v -, v -, v 6-9, v -7 - ( pt) Library/TCNJ/TCNJ VectorEquations/problem8pg Find the value of a for which v a 8 is in the set H span a,, ( pt) Library/TCNJ/TCNJ VectorSpaces/problempg Let H be the set of all vectors of the form: t Find a -t vector v in R such that H Span{v} v where a is in R Let W be the set of all polynomials of the form p(t) t + a, where a is in R Let W be the set of all polynomials of the form p(t) at + at, where a is in R ( pt) tcnjlibrary/tcnj VectorEquations/problempg Let x,y,z be vectors and let w x + y + z If z x+y, then w is a linear combination of x and y We have w x+ y Based upon the above calculation, exactly of the following statements must be true Determine these true statments Hint: If Span(x,w)Span(x,y,z) is true, then every linear combination of the vectors x and w can be rewritten as a linear combination of the vectors x,y,z and conversely every linear combination of the vectors x,y,z can be rewritten as a linear combination of the vectors x,w A Span(x,w) Span(x,y,z) B Span(x,z) Span(w,z) C Span(x,y,z) Span(w,x,z) D Span(x,y) Span(x,y,z) ( pt) tcnjlibrary/tcnj VectorEquations/problempg Let A and b ? Is b a linear combination of the columns of A? If b is a linear combination, find weights such that a + a + a b, ( pt) Library/TCNJ/TCNJ VectorSpaces/problempg Let W be the set of all vectors of the form: b+c b c vectors u and v suchthat W Span{u,v} u, v Find where a, a, a are the three columns of A If b is not a linear combination, enter s for the coefficients 6 ( pt) tcnjlibrary/tcnj VectorEquations/problempg For each set {u,v} of vectors, determine whether Span(u,v) is a line or a plane ( pt) Library/TCNJ/TCNJ VectorSpaces/problem6pg Determine if each of the following sets is a subspace of P n, for an appropriate value of n Type yes or no for each answer Let W be the set of all polynomials of the form p(t) at,? u - 8-8? u -, v,, v 6, -9

16 ? u -, v 6 -, 6? u - - -, v 6 7, 7 ( pt) tcnjlibrary/tcnj VectorEquations/problem7pg Let A 8 and b 8 Denote the columns of A by a, a, a? Determine if b is in the set {a,a,a }? Determine if b is in the set Span(a,a,a ) How many vectors are in {a,a,a }? (For infinitely many, enter -) How many vectors are in Span(a,a,a )? (For infinitely many, enter -) 8 ( pt) tcnjlibrary/tcnj VectorEquations/problem9pg Find the value of a for which v 8 a is in the set H span a,, Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

17 Ivan Ivanov WeBWorK assignment number hw6 is due : /6/ at :pm EST The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information linalg This file is /conf/snippets/setheaderpg you can use it as a model for creating files which introduce each problem set The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help Don t spend a lot of time guessing it s not very efficient or effective Give or significant digits for (floating point) numerical answers For most problems when entering numerical answers, you can if you wish enter elementary expressions such as instead of 8, sin( pi/)instead of -, e (ln()) instead of, ( +tan()) ( sin()) 6 7/8 instead of 76, etc Here s the list of the functions which WeBWorK understands You can use the Feedback button on each problem page to send to the professors ( pt) rochesterlibrary/setlinearalgebra9dependence/ur la 9 pg - Let A - -, B -, C -8 -, and D - - -? Determine whether or not the four vectors listed above are linearly independent or linearly dependent If they are linearly dependent, determine a non-trivial linear relation - (a non-trivial relation is three numbers which are not all three zero) Otherwise, if the vectors are linearly independent, enter s for the coefficients, since that relationship always holds A+ B+ C+ D You can use this row reduction tool to help with the calculations ( pt) rochesterlibrary/setlinearalgebra9dependence/ur la 9 7pg Thevectors v - -, u -, and w k 6 are linearly independent if and only if k ( pt) rochesterlibrary/setlinearalgebra9dependence- /ur la 9 pg Find a linearly independent set of vectors that spans the same subspace of R as that spanned by the vectors -, - -, Linearly independent set:, ( pt) rochesterlibrary/setlinearalgebra9dependence- /ur la 9 pg Find a linearly independent set of vectors that spans the same subspace of R as that spanned by the vectors - -, - -, -7 - Linearly independent set:, - -9, ( pt) rochesterlibrary/setlinearalgebrabases/ur la pg The vectors v -, v -7, and v - 6 k form a basis for R if and only if k 6 ( pt) rochesterlibrary/setlinearalgebrabases/ur la pg Consider the basis B of R consisting of vectors - -7 and -6 Find x in R whose coordinate vector relative to the basis B is - x B x 7 ( pt) rochesterlibrary/setlinearalgebrabases/ur { } la pg 8 The set B, is a basis for R 8 Find the coordinates of the vector x relative to the 6 basis B: x B

18 8 ( pt) rochesterlibrary/setlinearalgebrabases/ur la pg Find the coordinate vector of x with respect to the basis B,, or R x B 9 ( pt) rochesterlibrary/setlinearalgebrabases/ur la 8pg -6 Let A - Find bases of the kernel and image of A (or the linear transformation T (x) Ax) Kernel: Image:, A - - -, ( pt) rochesterlibrary/setlinearalgebra7dotproductrn- /ur la 7 7pg Find the value of k for which the vectors x - k and y - - k are orthogonal ( pt) rochesterlibrary/setlinearalgebra7dotproductrn- /ur la 7 pg Find two linearly independent vectors perpendicular to the vector v - 7-7, ( pt) rochesterlibrary/setlinearalgebrabases/ur la 8pg Find bases of the kernel and image of the orthogonal projection onto the plane x y + z in R Kernel: Image:, ( pt) rochesterlibrary/setlinearalgebrabases/ur la pg Find a basis of the subspace of R spanned by the following vectors: 6 6 -, -, -, , ( pt) rochesterlibrary/setlinearalgebrabases/ur la pg Find a basis of the subspace of R defined by the equation 6x x 8x, ( pt) rochesterlibrary/setlinearalgebrabases/ur la pg Find a basis of the column space of the matrix 6 ( pt) rochesterlibrary/setlinearalgebra7dotproductrn- /ur la 7 pg - Let v, v -, and v - - Find a vector v in R such that the vectors v, v, v, and v are orthonormal v 7 ( pt) rochesterlibrary/setlinearalgebra8orthogonalbases- /ur la 8 pg Perform the Gram-Schmidt process on the following sequence of vectors to obtain ortonormal basis x , y, 6, z - 6, 8 ( pt) rochesterlibrary/setlinearalgebra8orthogonalbases- /ur la 8 6pg Find an orthonormal basis of the plane x 6x x,

19 9 ( pt) rochesterlibrary/setlinearalgebra8orthogonalbases- /ur la 8 7pg Let A Find an orthonormal basis of the kernel of A, Let x -, y - -7, and z Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R spanned by x, y, and z,, ( pt) rochesterlibrary/setlinearalgebra8orthogonalbases- /ur la 8 pg Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester