MAT 242 Test 2 SOLUTIONS, FORM T


 Scot Warren Wells
 1 years ago
 Views:
Transcription
1 MAT 242 Test 2 SOLUTIONS, FORM T Let v =, v 5 2 =, v 3 =, and v 5 4 = a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these vectors that adds up to. To do this, you need to find a nontrivial solution to c v + c 2 v 2 + c 3 v 3 + c 4 v 4 =, whose matrix looks like [ v v 2 v 3 v 4 ], or RREF The nontrivial solutions are thus given by c = 2t c 2 = 5t c 3 = t c 4 = where t can be any real number Choosing t = produces the linear combination 2v 5v 2 + v 3. Grading: +5 points for setting up the system of linear equations; +5 points for parameterizing it; +5 points for finding a linear combination. Grading for common mistakes: +7 points (total) for using the column space method instead. 4 3 b. [ points] Is the vector in the span of { v 6, v 2, v 3, v 4 }? Justify your answer. 5 You must determine whether the augmented matrix [ v v 2 v 3 v 4 u] is a system that has at least one solution RREF Since this system has no solutions, the vector is not in the span. Grading: +4 points for setting up the matrix, +4 points for the RREF, +3 points for determining how many solutions there were, +4 points for answering NO. Grading for common mistakes: 8 points for using instead of u. 2
2 MAT 242 Test 2 SOLUTIONS, FORM T 2. [25 points] For the matrix A below, find a basis for the null space of A, a basis for the row space of A, a basis for the column space of A, the rank of A, and the nullity of A. The reduced row echelon form of A is the matrix R given below A = R = To find a basis for the null space, you need to solve the system of linear equations A x =, or equivalently, R x =. Parameterizing the solutions to this equation, and writing the answer in expanded form, produces x x 3 = α + α 2 so x 4 x , is a basis for the null space of A. The nullity is the number of vectors in this basis, namely 2. A basis for the row space can be found by taking the nonzero rows of R: {[,, 5, 2, ], [,, 3,, ], [,,,, ]} A basis for the column space can be found by taking the columns of A which have pivots in them, so ,, is a basis for the column space of A. Lastly, the rank of A is the number of vectors in a basis for the row space (or column space) of A, so the rank of A is 3. Grading: +5 points for each of: finding a basis for the null space, a basis for the row space, a basis for the column space, the nullity, and the rank. Grading for common mistakes: 3 points for forgetting a variable in the parameterization; 3 points for choosing columns of R for the column space of A; 3 points for choosing rows from A for the row space of A; 3 points for choosing the nonpivot columns of A for the null space of A. 3
3 MAT 242 Test 2 SOLUTIONS, FORM T [5 points] Find the eigenvalues of A = Solve for when the determinant of A λi equals zero: 6 λ λ λ = λ3 + λ 2 6λ = λ (λ + 3) (λ 2). The eigenvalues make this product equal to zero, so they are, 3, and 2. Grading: +5 points for A λi, +5 points for finding the determinant, +5 points for finding the eigenvalues. Grading for common mistakes: +5 points (total) for using row operations on the matrix A, +3 points (total) for finding the eigenvectors of. 4. Let B be the (ordered) basis 2, 2 5, 3 9 and C the basis, 3, a. [ points] Find the coordinates of 6 8 with respect to the basis C. 38 [ u] C = C u = = Grading: +3 points for the formula, +4 points for substitution, +3 points for calculation. b. [ points] Find the changeofbasis matrix from C to B. 2 3 B C = = Grading: +3 points for the formula, +4 points for substitution, +3 points for calculation Grading for common mistakes: +7 points (total) for C B = (backwards), +5 points (total) for BC = 2 2, +5 points (total) for CB = 3 2 4, +3 points (total) for B = , +7 points (total) for B C x 396 6, +3 points (total) for any other 282 vector. 4
4 MAT 242 Test 2 SOLUTIONS, FORM T 5. The eigenvalues of the matrix A = are (with multiplicity 2) and 2 (with multiplicity ). 2 2 (You do not need to find these.) Do the following for the matrix A: a. [ points] Find a basis for the eigenspace of each eigenvalue. The eigenspace of an eigenvalue λ is the null space of A λi. So, if λ =, A ()I = RREF, 2 2 whose solutions (with a vector of s on the righthand side) are parameterized by and a basis for the eigenspace of is If λ = 2, whose solutions are parameterized by and a basis for the eigenspace of 2 is x = s + t x 3,. A (2)I = RREF 3 2, 2 4 x = s 3 2 x Grading: +3 points for A λi, +3 points for the RREF, +4 points for finding the null space basis. Grading for common mistakes: +5 points (total) for finding the column space of A lambdai. b. [ points] Is the matrix A diagonalizable? If so, find matrices D and P such that A = P DP and D is a diagonal matrix. If A is not diagonalizable explain carefully why it is not. YES. The dimension of each eigenspace equals the (given) multiplicity of each eigenvalue. The matrix P can be constructed by gluing the basis vectors together; then the (i, i) entry in D is the eigenvalue that the ith column of P belongs to. One pair of matrices that diagonalizes A is D = and P = Grading: +3 points for YES, +7 points for P and D. Full credit marked + points was given for an answer consistent with any mistakes made in part (a). 5
5 MAT 242 Test 2 SOLUTIONS, FORM W Let S =,,,, a. [ points] Find a basis for the subspace W spanned by S, and the dimension of W. To find a basis, use the column space (CS) approach: Glue the vectors together to get the matrix Ṽ, find the RREF, and take the original vectors that have pivots in their columns: Ṽ = 2 RREF A basis for the subspace will consist of the columns of Ṽ which have a pivot in their column, namely ,,. The dimension of the subspace is the number of vectors in this basis, which is 3. Grading: +5 points for Ṽ, +5 points for the RREF, +5 points for the dimension. Grading for common mistakes: +5 points (total) for finding a basis for the null space; 3 points for using the columns of the RREF. 7 b. [ points] Is the vector in the span of S? Justify your answer. 4 3 You must determine whether the augmented matrix [ v v 2 v 3 v 4 u] is a system that has at least one solution RREF Since this system has infinitely many solutions, it has at least one, and the vector is in the span. Grading: +4 points for setting up the matrix, +4 points for the RREF, +3 points for determining how many solutions there were, +4 points for answering YES. Grading for common mistakes: 8 points for using instead of u. 2
6 MAT 242 Test 2 SOLUTIONS, FORM W 2. [25 points] For the matrix A below, find a basis for the null space of A, a basis for the row space of A, a basis for the column space of A, the rank of A, and the nullity of A. The reduced row echelon form of A is the matrix R given below A = R = To find a basis for the null space, you need to solve the system of linear equations A x =, or equivalently, R x =. Parameterizing the solutions to this equation, and writing the answer in expanded form, produces 5 so x 3 = α x + α 3 2 x , 5 is a basis for the null space of A. The nullity is the number of vectors in this basis, namely 2. A basis for the row space can be found by taking the nonzero rows of R: {[,,, 5 ], [,, 3, 5 ]} A basis for the column space can be found by taking the columns of A which have pivots in them, so , is a basis for the column space of A. Lastly, the rank of A is the number of vectors in a basis for the row space (or column space) of A, so the rank of A is 2. Grading: +5 points for each of: finding a basis for the null space, a basis for the row space, a basis for the column space, the nullity, and the rank. Grading for common mistakes: 3 points for forgetting a variable in the parameterization; 3 points for choosing columns of R for the column space of A; 3 points for choosing rows from A for the row space of A; 3 points for choosing the nonpivot columns of A for the null space of A. 3
7 MAT 242 Test 2 SOLUTIONS, FORM W 3. [5 points] Find the eigenvalues of A = Solve for when the determinant of A λi equals zero: λ 3 3 λ 3 λ = λ3 + 3λ 2 = λ 2 (λ + 3). The eigenvalues make this product equal to zero, so they are,, and 3. Grading: +5 points for A λi, +5 points for finding the determinant, +5 points for finding the eigenvalues. Grading for common mistakes: +5 points (total) for using row operations on the matrix A, +3 points (total) for finding the eigenvectors of. 4. Let B be the (ordered) basis 3, 2 7, 2 and C the basis, 3, a. [ points] Find the coordinates of 9 with respect to the basis B. 23 [ u] B = B 2 u = = Grading: +3 points for the formula, +4 points for substitution, +3 points for calculation. b. [ points] If the coordinates of u with respect to B are 6, what are the coordinates of u with 2 respect to C? [ u] C = C B [ u] B = = Grading: +3 points for the formula, +4 points for substitution, +3 points for calculation. Grading for common mistakes: +7 points (total) for B C[ u] = (backwards), +5 points 9 (total) for CB [ u] = , +5 points (total) for BC [ u] = 7, +3 points (total) for C [ u] = , +7 points (total) for the changeofbasis matrix C B = 3 7 2, points (total) for a different changeofbasis matrix. 4
8 MAT 242 Test 2 SOLUTIONS, FORM W 5. The eigenvalues of the matrix A = 3 3 are 3 (with multiplicity 2) and 4 (with multiplicity ). (You do not need to find these.) Do the following for the matrix 4 A: a. [ points] Find a basis for the eigenspace of each eigenvalue. The eigenspace of an eigenvalue λ is the null space of A λi. So, if λ = 3, A ( 3)I = RREF, whose solutions (with a vector of s on the righthand side) are parameterized by and a basis for the eigenspace of 3 is If λ = 4, whose solutions are parameterized by and a basis for the eigenspace of 4 is x = s + t x 3,. A ( 4)I = RREF, x = s x 3.. Grading: +3 points for A λi, +3 points for the RREF, +4 points for finding the null space basis. Grading for common mistakes: +5 points (total) for finding the column space of A lambdai. b. [ points] Is the matrix A diagonalizable? If so, find matrices D and P such that A = P DP and D is a diagonal matrix. If A is not diagonalizable explain carefully why it is not. YES. The dimension of each eigenspace equals the (given) multiplicity of each eigenvalue. The matrix P can be constructed by gluing the basis vectors together; then the (i, i) entry in D is the eigenvalue that the ith column of P belongs to. One pair of matrices that diagonalizes A is D = 3 3 and P = 4 Grading: +3 points for YES, +7 points for P and D. Full credit marked + points was given for an answer consistent with any mistakes made in part (a). 5
9 MAT 242 Test 2 SOLUTIONS, FORM O Let S =,,,, a. [ points] Find a basis for the subspace W spanned by S, and the dimension of W. To find a basis, use the column space (CS) approach: Glue the vectors together to get the matrix Ṽ, find the RREF, and take the original vectors that have pivots in their columns: Ṽ = 4 2 RREF A basis for the subspace will consist of the columns of Ṽ which have a pivot in their column, namely 3 4 3,,. The dimension of the subspace is the number of vectors in this basis, which is 3. Grading: +5 points for Ṽ, +5 points for the RREF, +5 points for the dimension. Grading for common mistakes: +5 points (total) for finding a basis for the null space; 3 points for using the columns of the RREF. 65 b. [ points] Is the vector in the span of S? Justify your answer You must determine whether the augmented matrix [ v v 2 v 3 v 4 u] is a system that has at least one solution RREF Since this system has no solutions, the vector is not in the span. Grading: +4 points for setting up the matrix, +4 points for the RREF, +3 points for determining how many solutions there were, +4 points for answering NO. Grading for common mistakes: 8 points for using instead of u. 2
10 MAT 242 Test 2 SOLUTIONS, FORM O 2. [25 points] For the matrix A below, find a basis for the null space of A, a basis for the row space of A, a basis for the column space of A, the rank of A, and the nullity of A. The reduced row echelon form of A is the matrix R given below A = R = To find a basis for the null space, you need to solve the system of linear equations A x =, or equivalently, R x =. Parameterizing the solutions to this equation, and writing the answer in expanded form, produces x x 3 = α + α 2 so x 4 x , is a basis for the null space of A. The nullity is the number of vectors in this basis, namely 2. A basis for the row space can be found by taking the nonzero rows of R: {[,,, 4, 3 ], [,,,, 5 ], [,,,, ]} A basis for the column space can be found by taking the columns of A which have pivots in them, so , 4, is a basis for the column space of A. Lastly, the rank of A is the number of vectors in a basis for the row space (or column space) of A, so the rank of A is 3. Grading: +5 points for each of: finding a basis for the null space, a basis for the row space, a basis for the column space, the nullity, and the rank. Grading for common mistakes: 3 points for forgetting a variable in the parameterization; 3 points for choosing columns of R for the column space of A; 3 points for choosing rows from A for the row space of A; 3 points for choosing the nonpivot columns of A for the null space of A. 3
11 MAT 242 Test 2 SOLUTIONS, FORM O 3. [5 points] Find the eigenvalues of A = Solve for when the determinant of A λi equals zero: λ 4 2 λ 2 2 λ = λ3 4λ = λ (λ + 2) (λ 2). The eigenvalues make this product equal to zero, so they are, 2, and 2. Grading: +5 points for A λi, +5 points for finding the determinant, +5 points for finding the eigenvalues. Grading for common mistakes: +5 points (total) for using row operations on the matrix A, +3 points (total) for finding the eigenvectors of. 4. Let B be the (ordered) basis,, 2 3 and C the basis 3, 2, a. [ points] Find the coordinates of 3 38 with respect to the basis C. 8 [ u] C = C u = = Grading: +3 points for the formula, +4 points for substitution, +3 points for calculation. b. [ points] If the coordinates of u with respect to B are 8, what are the coordinates of u with respect to C? [ u] C = C B [ u] B = = Grading: +3 points for the formula, +4 points for substitution, +3 points for calculation. Grading for common mistakes: +7 points (total) for B C[ u] = (backwards), +5 points 6 (total) for CB [ u] = , +5 points (total) for BC [ u] = 2, +3 points (total) for C [ u] = , +7 points (total) for the changeofbasis matrix C B = 2 7, points (total) for a different changeofbasis matrix. 4
12 MAT 242 Test 2 SOLUTIONS, FORM O The eigenvalues of the matrix A = 3 6 are 5 (with multiplicity 2) and 3 (with multiplicity ). (You do not need to find these.) Do the following for the matrix 8 A: a. [ points] Find a basis for the eigenspace of each eigenvalue. The eigenspace of an eigenvalue λ is the null space of A λi. So, if λ = 5, 6 32 A ( 5)I = 8 6 RREF 2, 8 6 whose solutions (with a vector of s on the righthand side) are parameterized by and a basis for the eigenspace of 5 is If λ = 3, whose solutions are parameterized by and a basis for the eigenspace of 3 is x = s + t 2 x 3, A (3)I = 6 6 RREF 2, 8 8 x = s 2 x Grading: +3 points for A λi, +3 points for the RREF, +4 points for finding the null space basis. Grading for common mistakes: +5 points (total) for finding the column space of A lambdai. b. [ points] Is the matrix A diagonalizable? If so, find matrices D and P such that A = P DP and D is a diagonal matrix. If A is not diagonalizable explain carefully why it is not. YES. The dimension of each eigenspace equals the (given) multiplicity of each eigenvalue. The matrix P can be constructed by gluing the basis vectors together; then the (i, i) entry in D is the eigenvalue that the ith column of P belongs to. One pair of matrices that diagonalizes A is D = 5 5 and P = Grading: +3 points for YES, +7 points for P and D. Full credit marked + points was given for an answer consistent with any mistakes made in part (a). 5
Similarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationUniversity of Ottawa
University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Alistair Savage Final Exam April 2013 Surname First Name Student # Seat # Instructions: (a)
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationMATH 551  APPLIED MATRIX THEORY
MATH 55  APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationMAT 242 Test 3 SOLUTIONS, FORM A
MAT Test SOLUTIONS, FORM A. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. A = 8 a. [5 points] Find the orthogonal projection
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationMATH10212 Linear Algebra B Homework 7
MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments
More informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationLecture Notes 2: Matrices as Systems of Linear Equations
2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably
More informationSection 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =
Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationMath 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns
Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in
More informationLecture Note on Linear Algebra 15. Dimension and Rank
Lecture Note on Linear Algebra 15. Dimension and Rank WeiShi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationReduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:
Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in
More informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero
More informationExamination paper for TMA4115 Matematikk 3
Department of Mathematical Sciences Examination paper for TMA45 Matematikk 3 Academic contact during examination: Antoine Julien a, Alexander Schmeding b, Gereon Quick c Phone: a 73 59 77 82, b 40 53 99
More informationMath 240: Linear Systems and Rank of a Matrix
Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011
More information1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0
Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are
More informationMA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam
MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am  :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.
More informationBasic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.
Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required
More informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More information1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)
Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationx + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3
Math 24 FINAL EXAM (2/9/9  SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r
More informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for inclass presentation
More information4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
More information4 Solving Systems of Equations by Reducing Matrices
Math 15 Sec S0601/S060 4 Solving Systems of Equations by Reducing Matrices 4.1 Introduction One of the main applications of matrix methods is the solution of systems of linear equations. Consider for example
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationJones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION
8498_CH08_WilliamsA.qxd 11/13/09 10:35 AM Page 347 Jones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION C H A P T E R Numerical Methods 8 I n this chapter we look at numerical techniques for
More informationINTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationSTUDY GUIDE LINEAR ALGEBRA. David C. Lay University of Maryland College Park AND ITS APPLICATIONS THIRD EDITION UPDATE
STUDY GUIDE LINEAR ALGEBRA AND ITS APPLICATIONS THIRD EDITION UPDATE David C. Lay University of Maryland College Park Copyright 2006 Pearson AddisonWesley. All rights reserved. Reproduced by Pearson AddisonWesley
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationSergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014
Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of
More information( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&
Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important
More informationMath Practice Problems for Test 1
Math 290  Practice Problems for Test 1 UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT. 3 4 5 1. Let c 1 and c 2 be the columns of A 5 2 and b 1. Show that b Span{c 1, c 2 } by 6 6 6 writing b as a linear
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationRow Echelon Form and Reduced Row Echelon Form
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for inclass presentation
More informationPresentation 3: Eigenvalues and Eigenvectors of a Matrix
Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationSolution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.
Solutions to Math 30 Takehome prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationSALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated
SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT240 Action Taken (Please Check One) New Course Initiated
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More information4.1 VECTOR SPACES AND SUBSPACES
4.1 VECTOR SPACES AND SUBSPACES What is a vector space? (pg 229) A vector space is a nonempty set, V, of vectors together with two operations; addition and scalar multiplication which satisfies the following
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationsome algebra prelim solutions
some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationLecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25Sep02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
More informationSection 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)
Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix
More informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationCHARACTERISTIC ROOTS AND VECTORS
CHARACTERISTIC ROOTS AND VECTORS 1 DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 11 Statement of the characteristic root problem Find values of a scalar λ for which there exist vectors x 0 satisfying
More informationProblems for Advanced Linear Algebra Fall 2012
Problems for Advanced Linear Algebra Fall 2012 Class will be structured around students presenting complete solutions to the problems in this handout. Please only agree to come to the board when you are
More informationMatrices in Statics and Mechanics
Matrices in Statics and Mechanics Casey Pearson 3/19/2012 Abstract The goal of this project is to show how linear algebra can be used to solve complex, multivariable statics problems as well as illustrate
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationVector Spaces 4.4 Spanning and Independence
Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationChapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an
More informationAbstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
More informationNotes on Jordan Canonical Form
Notes on Jordan Canonical Form Eric Klavins University of Washington 8 Jordan blocks and Jordan form A Jordan Block of size m and value λ is a matrix J m (λ) having the value λ repeated along the main
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationTv = Tu 1 + + Tu N. (u 1,...,u M, v 1,...,v N ) T(Sv) = S(Tv) = Sλv = λ(sv).
54 CHAPTER 5. EIGENVALUES AND EIGENVECTORS 5.6 Solutions 5.1 Suppose T L(V). Prove that if U 1,...,U M are subspaces of V invariant under T, then U 1 + + U M is invariant under T. Suppose v = u 1 + + u
More informationMATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,
MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationAPPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A.
APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the cofactor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationPractice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.
Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationThe Hadamard Product
The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationC 1 x(t) = e ta C = e C n. 2! A2 + t3
Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix
More informationMatrices, Determinants and Linear Systems
September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More information