MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018
|
|
- Beryl Ross
- 5 years ago
- Views:
Transcription
1 Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry and row Free variable Row operations, Gaussian elimination Row echelon form, Reduced row echelon form Vectors Matrix addition, subtraction, multiplication, scalar multiplication Symmetric matrix Matrix transpose Identity matrix Singular, nonsingular Matrix inverse, invertible matrices Determinants On homework or exams I expect you to tell me which row operations you are using. following notation can be used to state a row operation: The Interchange rows 1 and 3 R 1 R 3 Multiply row 2 by -7 7R 2 R 2 Replace row 3 by the sum of row 3 and 5 times row 4 5R 4 + R 3 R 3 This notation tells us which rows were used and which rows were ultimately changed. 1. Solve the system using row operations to put the associated matrix in strictly triangular form and then back substitute. 3x 1 + 2x 2 + x 3 = 0 2x 1 + x 2 x 3 = 2 2x 1 x 2 + 2x 3 = 1 1
2 2. Which of the following matrices are in row echelon form? Which are in reduced row echelon form? [ ] (b) (d) (e) (f) (g) (h) For each of the systems of equations that follow, use Gaussian elimination to obtain a solution involving reduced row echelon form. If there are infinite solutions, identify the free variables and describe all solutions. (b) x 1 2x 2 = 3 2x 1 x 2 = 9 2x 1 + 3x 2 + x 3 = 1 x 1 + x 2 + x 3 = 3 3x 1 + 4x 2 + 2x 3 = 4 x 1 2x 2 = 3 2x 1 + x 2 = 1 5x 1 + 8x 2 = 4 2
3 4. If compute the following quantities A = and B = 3 1 1, A (b) A + B 2A 3B (d) (2A) T (3B) T (e) AB (f) BA (g) A T B T (h) (BA) T 5. For each pair of matrices, determine whether it is possible to perform the multiplication. If it is possible, compute the product. [ ] (b) [ ] (d) [ ] [ ] (e) [ ] [ ] (f) 2 1 [ ] 3 6. Let A and B be symmetric n n matrices. For each of the following, determine whether the given matrix will be symmetric or non-symmetric. It will be helpful to know that the (i, j)-entry of a matrix M = AB, denoted m ij, can be computed with the sum n m ij = a ik b kj, k=1 where A is an m n matrix, B is an n r matrix, and M is an m r matrix. Another hint is that A T = A, or that a ij = a ji. C = A + B (b) D = A 2 E = AB (d) F = ABA (e) G = AB + BA (f) H = AB BA 3
4 7. Let C be a non-symmetric n n matrix. For each of the following, determine whether the given matrix must be symmetric or could be non-symmetric. A = C + C T (b) B = C C T D = C T C (d) E = C T C CC T (e) F = (I + C)(I + C T ) (f) G = (I + C)(I C T ) 8. Evaluate the following determinants by hand (b) (d) (e) (f) Let A and B be 2 2 matrices. Justify your answers by either proving the statement is true or by providing a counterexample. Does det (A + B) = det (A) + det (B)? (b) Does det (AB) = det (A) det (B)? Does det (AB)) = det (BA)? (d) Does det (A T ) = det (A)? 4
5 Homework #2 Assigned: August 24, Let V = R, so the vectors are real numbers. To remind us that vectors are real numbers in this example, feel free to not use vector notation (use x instead of x). Define scalar multiplication as usual αx = α x, and define addition, denoted by, by x y = max {x, y}. Is V a vector space? Clearly spell out why or why not (every property needs explanation). Assume nothing is clear to the reader and everything needs to be explained. Also, use the names of the properties you are checking along with the numbers we use in class for the sake of clarity. Since you need the practice, make sure you address all ten properties. 2. Determine whether the following sets form a subspace of R 3. {[x 1, x 2, x 3 ] T : x 1 + x 3 = 1} (b) {[x 1, x 2, x 3 ] T : x 1 = x 2 = x 3 } 3. Determine whether the following sets form a subspace of R 2 2. The set of all 2 2 lower triangular matrices. You may need to find out what this means. Try (b) The set of all 2 2 matrices A such that a 11 = 1. The set of all 2 2 matrices B such that b 11 = 0. (d) The set of all symmetric 2 2 matrices. (e) The set of all singular 2 2 matrices. The following theorem will be helpful. Theorem 1. An n n matrix A is singular if and only if det (A) = Determine whether the following sets form a subspace of P 4. The set of all polynomials in P 4 of even degree. (b) The set of all polynomials p(x) P 4 such that p(0) = Determine whether the following sets form a subspace of C[ 1, 1]. The set of odd functions in C[ 1, 1]. Recall that a function is odd if f( x) = f(x) for all x in its domain, which is [ 1, 1]. (b) The set of functions in C[ 1, 1] such that f( 1) = 0 and f(1) = 0. The set of functions in C[ 1, 1] such that f( 1) = 0 or f(1) = 0. 5
6 6. Let V be a vector space and let x, y, z V. Prove the following: If x + y = x + z, then y = z. (b) β0 = 0 for every β R. Use the fact that β0 = β(0 + 0). If αx = 0 and α is nonzero, then x = 0. Use (b). 6
7 Homework #3 Assigned: September 7, Let A R n n, and define the set C(A) to be the set of all matrices that commute with A, or Show C(A) is a subspace of R n n. C(A) = {B : B R n n and AB = BA}. 2. In each of the following, determine the subspace of R 2 2 consisting of all matrices that commute with the given matrix. As an example, which you might want to work out for yourself so that you understand the process before trying the other problems, consider ([ ]) {[ ] } 1 0 b11 0 C = : b b 11, b 22 R. 22 [ ] [ ] 1 1 (b) J = 0 1 [ ] Let A R 2 2. Determine whether the following sets are subspaces of R 2 2. S 1 = {B R 2 2 : BA = O}, where O is the matrix of all zeros. (b) S 2 = {B R 2 2 : AB BA} S 3 = {B R 2 2 : AB + B = O} 4. Given 1 3 x 1 = 2 x 2 = u = 6 v = 2, 6 5 answer the following questions. Justify your answers by writing out the necessary linear combination or explaining why no linear combination is possible. Is u Span (x 1, x 2 )? (b) Is v Span (x 1, x 2 )? 7
8 5. Determine whether the following are spanning sets for R 2. Justify your answers. {[ ] [ ]} 2 4, 3 6 (b) {[ ] 2, 1 [ ] 1, 3 [ ]} Determine whether the following sets are spanning sets for P 3. Justify your answers. {1, x 2, x 2 2} (b) {x + 2, x + 1, x 2 1} 8
9 Homework #4 Assigned: September 17, 2018 If you find the need to row reduce a matrix, feel free to use technology to do so. If you do use technology, report your work like I do in class: state the matrix to be reduced and its final reduced state, but do not feel the need to provide the intermediate steps. 1. Determine whether the following vectors are linearly independent in R 3. (b) , 1, , 1, , Determine whether the following vectors are linearly independent in P 3. x + 2, x + 1, x 2 1 (b) x + 2, x Let x 1, x 2, and x 3 be linearly independent vectors in R n, and let y 1 = x 2 x 1, y 2 = x 3 x 2, y 3 = x 3 x 1. Are y 1, y 2, and y 3 linearly independent? Justify your answer. 4. Let A be an m n matrix. Prove that if A has linearly independent column vectors, then N(A) = {0}. Hint: for any x R n, Ax = x 1 a 1 + x 2 a x n a n, where a i is the i-th column of A. 5. Let S and T be subspaces of a vector space V. Prove that their intersection, S T = {v : v S and v T }, is a subspace of V. 9
10 Homework #5 Assigned: September 26, 2018 If you find the need to row reduce a matrix, feel free to use technology to do so. If you do use technology, report your work like I do in class: state the matrix to be reduced and its final reduced state, but do not feel the need to provide the intermediate steps. 1. Find a basis for the subspace S of R 4 consisting of all vectors of the form a + b a b + 2c b, c where a, b, c R. What is the dimension of S? Justify your answers. 2. The vectors x 1 = 2, x 2 = 5, x 3 = 3, x 4 = 7, x 5 = span R 3. Pare down the set {x 1, x 2, x 3, x 4, x 5 } to form a basis of R 3. Justify your answer. 3. Let S be the subspace of P 3 consisting of all polynomials p(x) such that p(0) = 0, and let T be the subspace of all polynomials q(x) such that q(1) = 0. Find bases for the following sets. S (b) T S T 4. Recall an early example of a vector space: {[ ] } x1 V = : x 1, x 2 R, x 2 > 0, x 2 where for α R and x, y V, [ ] αx1 α x = x α 2 [ ] x1 + y and x y = 1. x 2 y 2 Do the vectors [ ] [ ] 1 1, 2 4 form a basis for V? 10
11 Homework #6 Assigned: October 19, Consider the following bases for R 2 : E = {e 1, e 2 }, U = {u 1, u 2 }, and W = {w 1, w 2 }, where u 1 = [ ] 1, u 2 2 = Find the transition matrix from U to E. (b) Find the transition matrix from E to U. [ ] 3 Convert x = to [x] 2 U. [ ] 2, w 5 1 = (d) Find the transition matrix from W to U. [ ] 1 (e) Convert [x] W = to [x] 4 U. 2. Given W v 1 = [ ] 2, v 6 2 = [ ] 3, w 2 2 = [ ] 1, and S 4 VU = [ ] 4. 3 [ ] find vectors u 1 and u 2 such that S is the transition matrix from V = {v 1, v 2 } to U = {u 1, u 2 }. 3. Let E = {1, x} and F = {2x 1, 2x + 1} be bases for P 2. Find the transition matrix from F to E. (b) Find the transition matrix from E to F. [ ] 3 Convert x = to [x] 1 E. Check that your answer is correct. F 4. Several collections of vectors exist that are used as bases for various vector spaces involving functions. Two such collections are the Laguerre polynomials, L, and the Hermite polynomials, H. Below are the first four polynomials for each collection, which each form a basis for P 4. L = {1, 1 t, 2 4t + t 2, 6 18t + 9t 2 t 3 } H = {1, 2t, 2 + 4t 2, 12t + 8t 3 } Prove that both of these collections form a basis for P 4. (b) Find the transition matrix from L to H. Convert 3 + t 6t 2 to both L and H. Note that the given polynomial is written in the standard basis for P 4. 11
12 Homework #7 Assigned: October 29, Find a basis for the row space, column space, and null space for A. Provide dimensions for each space and verify that the dimensions of the row space and column space are equal. Also, verify the Rank-Nullity Theorem A = Let A be a 6 n matrix of rank r, and let b be a vector in R 6. For each pair of values r and n that follow, indicate the possibilities as to the number of solutions one could have for the linear system Ax = b. Explain your answers. n = 7, r = 5 (b) n = 7, r = 6 n = 5, r = 5 (d) n = 5, r = 4 3. Determine whether the following are linear transformations. [ ] 1 + L 1 : R 3 R 2 x1, where L 1 (x) =. [ ] (b) L 2 : R 3 R 2 x, where L 2 (x) = 3. x 1 + x 2 x 2 L 3 : R n n R n n, where L 3 (A) = A T. (d) L 4 : R n n R n n, where L 4 (A) = A A T. 4. Let C be a fixed n n matrix. Determine whether the following are linear transformations from R n n to R n n. L 5 (A) = CA + AC (b) L 6 (A) = C 2 A L 7 (A) = A 2 C 5. Find the kernel and range of each of the following linear transformations from P 3 to P 3. L 8 (p(x)) = xp (x). (b) L 9 (p(x)) = p(x) p (x) L 10 (p(x)) = p(0)x + p(1). 6. Determine the kernel and range of each of the following linear transformations on R 3. L 11 (x) = [x 3, x 2, x 1 ] T 12
13 (b) L 12 (x) = [x 1, x 1, x 1 ] T 7. Let S be the subspace of R 3 spanned by e 1 and e 2. For each linear transformation in Problem 6, find L(S). 8. Let L : V W be a linear transformation, and let T be a subspace of W. The inverse image of T, denoted L 1 (T ), is defined by L 1 (T ) = {v V : L(v) T }. Show that L 1 (T ) is a subspace of V. (Note that L 1 (T ) is a set and that there is no inverse linear transformation L 1 : W V.) 9. For each of the following linear transformations L : R 3 R 2, find a matrix A such that L(x) = Ax for every x R 3. L 13 (x) = [x 1 + x 2, 0] T (b) L 14 (x) = [x 1, x 2 ] T L 15 (x) = [x 2 x 1, x 3 x 2 ] T 10. Find the standard matrix representation for each of the following linear transformations. L 16 is the linear transformation that rotates each x in R 2 by 45 in the clockwise direction. (b) L 17 is the linear transformation that reflects each vector x in R 2 about the x 1 axis and then rotates it 90 in the counterclockwise direction. L 18 is the linear transformation that doubles the length of x in R 2 and then rotates it 30 in the counterclockwise direction. (d) L 19 is the linear transformation that reflects each vector x in R 2 about the line x 2 = x 1 and then projects it onto the x 1 -axis. 13
14 Homework #8 Assigned: November 9, Find the eigenvalues and the corresponding eigenspaces for each of the following matrices. Show your work, though you may use technology to perform row reductions. Also state the algebraic and geometric multiplicities for each eigenvalue. (b) (d) A = B = [ ] [ ] C = D = Let A be an n n matrix, and let B = I 2A + A 2. Show that if x is an eigenvector of A belonging to an eigenvalue λ of A, then x is also an eigenvector of B belonging to an eigenvalue µ of B. How are λ and µ related? (b) Show that if λ = 1 is an eigenvalue of A, then the matrix B will be singular. 3. An n n matrix A is an idempotent if A 2 = A. Show that if λ is an eigenvalue of an idempotent matrix, then λ must be either 0 or Let A be a 2 2 matrix. If tr (A) = 8 and det A = 12, what are the eigenvalues of A? 5. Let A be an n n matrix, and let λ be an eigenvalue of A. Prove that E λ (A) is a subspace of R n. 6. Let A be an n n matrix and λ an eigenvalue of A. If A λi has rank k, what is the geometric multiplicity of E λ (A)? Explain your answer. 7. Diagonalize the following matrices, if possible. A = [ ]
15 (b) B = The city of Mawtookit maintains a constant population of 300,000 people from year to year. A political science study estimated that there were 150,000 Independents, 90,000 Democrats, and 60,000 Republicans in the town. It was also estimated that each year 20% of the Independents become Democrats and 10% become Republicans. Similarly, 20% of Democrats become Independents and 10% become Republicans. Finally, 10% of Republicans become Democrats and 10% become Independents. Find the transition matrix, M, for the Markov chain outlined above. (b) Diagonalize the matrix M found in. Which group will dominate town politics in the long run? Justify your answer. 15
MATH Linear Algebra Homework Solutions: #1 #6
MATH 35-0 Linear Algebra Homework Solutions: # #6 Homework # [Cochran-Bjerke, L Solve the system using row operations to put the associated matrix in strictly triangular form and then back substitute 3x
More information(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.
1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationMath 323 Exam 2 Sample Problems Solution Guide October 31, 2013
Math Exam Sample Problems Solution Guide October, Note that the following provides a guide to the solutions on the sample problems, but in some cases the complete solution would require more work or justification
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationInstructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences.
Instructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences.. Recall that P 3 denotes the vector space of polynomials of degree less
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More information(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).
.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)
More informationTMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013
TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)
More informationThe value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N.
Math 410 Homework Problems In the following pages you will find all of the homework problems for the semester. Homework should be written out neatly and stapled and turned in at the beginning of class
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationMATH 2360 REVIEW PROBLEMS
MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1
More informationMath 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level
More informationCalculating determinants for larger matrices
Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationFamily Feud Review. Linear Algebra. October 22, 2013
Review Linear Algebra October 22, 2013 Question 1 Let A and B be matrices. If AB is a 4 7 matrix, then determine the dimensions of A and B if A has 19 columns. Answer 1 Answer A is a 4 19 matrix, while
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More informationDepartment of Aerospace Engineering AE602 Mathematics for Aerospace Engineers Assignment No. 4
Department of Aerospace Engineering AE6 Mathematics for Aerospace Engineers Assignment No.. Decide whether or not the following vectors are linearly independent, by solving c v + c v + c 3 v 3 + c v :
More informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More informationProblem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show
MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationDaily Update. Math 290: Elementary Linear Algebra Fall 2018
Daily Update Math 90: Elementary Linear Algebra Fall 08 Lecture 7: Tuesday, December 4 After reviewing the definitions of a linear transformation, and the kernel and range of a linear transformation, we
More informationElementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th.
Elementary Linear Algebra Review for Exam Exam is Monday, November 6th. The exam will cover sections:.4,..4, 5. 5., 7., the class notes on Markov Models. You must be able to do each of the following. Section.4
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationspring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra
spring, 2016. math 204 (mitchell) list of theorems 1 Linear Systems THEOREM 1.0.1 (Theorem 1.1). Uniqueness of Reduced Row-Echelon Form THEOREM 1.0.2 (Theorem 1.2). Existence and Uniqueness Theorem THEOREM
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationCSL361 Problem set 4: Basic linear algebra
CSL361 Problem set 4: Basic linear algebra February 21, 2017 [Note:] If the numerical matrix computations turn out to be tedious, you may use the function rref in Matlab. 1 Row-reduced echelon matrices
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationMath 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures
More informationMATH 369 Linear Algebra
Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine
More informationLinear algebra I Homework #1 due Thursday, Oct Show that the diagonals of a square are orthogonal to one another.
Homework # due Thursday, Oct. 0. Show that the diagonals of a square are orthogonal to one another. Hint: Place the vertices of the square along the axes and then introduce coordinates. 2. Find the equation
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More information7.6 The Inverse of a Square Matrix
7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use Gauss-Jordan elimination to find inverses
More informationTopic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C
Topic 1 Quiz 1 text A reduced row-echelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correct-choice may have 0, 1, 2, or 3 choice may have 0,
More informationand let s calculate the image of some vectors under the transformation T.
Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationLINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationMath Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT
Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear
More informationMATH 225 Summer 2005 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 2005
MATH 225 Summer 25 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 25 Department of Mathematical and Statistical Sciences University of Alberta Question 1. [p 224. #2] The set of all
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationPOLI270 - Linear Algebra
POLI7 - Linear Algebra Septemer 8th Basics a x + a x +... + a n x n b () is the linear form where a, b are parameters and x n are variables. For a given equation such as x +x you only need a variable and
More informationSummer Session Practice Final Exam
Math 2F Summer Session 25 Practice Final Exam Time Limit: Hours Name (Print): Teaching Assistant This exam contains pages (including this cover page) and 9 problems. Check to see if any pages are missing.
More informationTest 3, Linear Algebra
Test 3, Linear Algebra Dr. Adam Graham-Squire, Fall 2017 Name: I pledge that I have neither given nor received any unauthorized assistance on this exam. (signature) DIRECTIONS 1. Don t panic. 2. Show all
More informationFinal Exam Practice Problems Answers Math 24 Winter 2012
Final Exam Practice Problems Answers Math 4 Winter 0 () The Jordan product of two n n matrices is defined as A B = (AB + BA), where the products inside the parentheses are standard matrix product. Is the
More informationSolutions to Final Exam
Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5-vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns
More informationRemark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
More informationis Use at most six elementary row operations. (Partial
MATH 235 SPRING 2 EXAM SOLUTIONS () (6 points) a) Show that the reduced row echelon form of the augmented matrix of the system x + + 2x 4 + x 5 = 3 x x 3 + x 4 + x 5 = 2 2x + 2x 3 2x 4 x 5 = 3 is. Use
More informationAnnouncements Monday, October 29
Announcements Monday, October 29 WeBWorK on determinents due on Wednesday at :59pm. The quiz on Friday covers 5., 5.2, 5.3. My office is Skiles 244 and Rabinoffice hours are: Mondays, 2 pm; Wednesdays,
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
More informationMAT 1302B Mathematical Methods II
MAT 1302B Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2015 Lecture 19 Alistair Savage (uottawa) MAT 1302B Mathematical Methods II Winter 2015 Lecture
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More information1 9/5 Matrices, vectors, and their applications
1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric
More information22A-2 SUMMER 2014 LECTURE 5
A- SUMMER 0 LECTURE 5 NATHANIEL GALLUP Agenda Elimination to the identity matrix Inverse matrices LU factorization Elimination to the identity matrix Previously, we have used elimination to get a system
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationWhat is A + B? What is A B? What is AB? What is BA? What is A 2? and B = QUESTION 2. What is the reduced row echelon matrix of A =
STUDENT S COMPANIONS IN BASIC MATH: THE ELEVENTH Matrix Reloaded by Block Buster Presumably you know the first part of matrix story, including its basic operations (addition and multiplication) and row
More informationMATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL
MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left
More informationEigenvalues and Eigenvectors A =
Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector
More informationCheat Sheet for MATH461
Cheat Sheet for MATH46 Here is the stuff you really need to remember for the exams Linear systems Ax = b Problem: We consider a linear system of m equations for n unknowns x,,x n : For a given matrix A
More informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationLinear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More information[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]
Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
More informationMath 265 Midterm 2 Review
Math 65 Midterm Review March 6, 06 Things you should be able to do This list is not meant to be ehaustive, but to remind you of things I may ask you to do on the eam. These are roughly in the order they
More informationMODULE 8 Topics: Null space, range, column space, row space and rank of a matrix
MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix Definition: Let L : V 1 V 2 be a linear operator. The null space N (L) of L is the subspace of V 1 defined by N (L) = {x
More informationLinear Algebra (MATH ) Spring 2011 Final Exam Practice Problem Solutions
Linear Algebra (MATH 4) Spring 2 Final Exam Practice Problem Solutions Instructions: Try the following on your own, then use the book and notes where you need help. Afterwards, check your solutions with
More informationMatrix Algebra. Matrix Algebra. Chapter 8 - S&B
Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number
More informationThe eigenvalues are the roots of the characteristic polynomial, det(a λi). We can compute
A. [ 3. Let A = 5 5 ]. Find all (complex) eigenvalues and eigenvectors of The eigenvalues are the roots of the characteristic polynomial, det(a λi). We can compute 3 λ A λi =, 5 5 λ from which det(a λi)
More informationLinear Algebra Exam 1 Spring 2007
Linear Algebra Exam 1 Spring 2007 March 15, 2007 Name: SOLUTION KEY (Total 55 points, plus 5 more for Pledged Assignment.) Honor Code Statement: Directions: Complete all problems. Justify all answers/solutions.
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More informationMath Computation Test 1 September 26 th, 2016 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge!
Math 5- Computation Test September 6 th, 6 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge! Name: Answer Key: Making Math Great Again Be sure to show your work!. (8 points) Consider the following
More informationRemark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called
More informationChapter 2 Notes, Linear Algebra 5e Lay
Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication
More informationPRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.
Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence
More information1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)
1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationDimension. Eigenvalue and eigenvector
Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,
More informationMath 314H Solutions to Homework # 3
Math 34H Solutions to Homework # 3 Complete the exercises from the second maple assignment which can be downloaded from my linear algebra course web page Attach printouts of your work on this problem to
More information4 Elementary matrices, continued
4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. To repeat the recipe: These matrices are constructed by performing the given row
More information