Coordinates. Definition Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let v be a vector in V, and write


 Joel Montgomery
 1 years ago
 Views:
Transcription
1 MATH10212 Linear Algebra Brief lecture notes 64 Coordinates Theorem 6.5 Let V be a vector space and let B be a basis for V. For every vector v in V, there is exactly one way to write v as a linear combination of the vectors in B. Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let v be a vector in V, and write v = c 1 v 1 + c 2 v c n v n Then c 1, c 2,..., c n are called the coordinates of v with respect to B, and the column vector c 1 c 2 [ v] B =. is called the coordinate vector of v with respect to B. c n Theorem 6.6 Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let u and v be vectors in V and let c be a scalar. Then a. [ u + v] B = [ u] B + [ v] B b. [c u] B = c[ u] B Theorem 6.7 Let B = { v 1, v 2,..., v n } be a basis for a vector space V and let u 1,..., u k be vectors in V. Then { u 1,..., u k } is linearly independent in V if and only if {[ u 1 ] B,..., [ u k ] B } is linearly independent in R n.
2 MATH10212 Linear Algebra Brief lecture notes 65 Dimension Theorem 6.8 Let B = { v 1, v 2,..., v n } be a basis for a vector space V. a. Any set of more than n vectors in V must be linearly dependent. b. Any set of fewer than n vectors in V cannot span V. Theorem 6.9 The Basis Theorem If a vector space V has a basis with n vectors, then every basis for V has exactly n vectors. A vector space V is called finitedimensional if it has a basis consisting of finitely many vectors. The dimension of V, denoted by dimv, is the number of vectors in a basis for V. The dimension of the zero vector space { 0} is defined to be zero. A vector space that has no finite basis is called infinitedimensional. Theorem 6.10 Let V be a vector space with dim V = n. Then a. Any linearly independent set in V contains at most n vectors. b. Any spanning set for V contains at least n vectors. c. Any linearly independent set of exactly n vectors in V is a basis for V. d. Any spanning set for V consisting of exactly n vectors is a basis for V. e. Any linearly independent set in V can be extended to a basis for V. f. Any spanning set for V can be reduced to a basis for V. Theorem 6.11 Let W be a subspace of a finitedimensional vector space V. Then a. W is finitedimensional and dimw dimv. b. dimw =dimv if and only if W = V.
3 MATH10212 Linear Algebra Brief lecture notes 66 Change of Basis ChangeofBasis Matrices Let B = { u 1,..., u n } and C = { v 1,..., v n } be bases for a vector space V. The n n matrix whose columns are the coordinate vectors [ u 1 ] C,..., [ u n ] C of the vectors in B with respect to C is denoted by P C B and is called the changeofbasis matrix from B to C. That is P C B = [[ u 1 ] C [ u 2 ] C... [ u n ] C ] Theorem 6.12 and Let be bases for a vector space V and let B = { u 1,..., u n } C = { v 1,..., v n } P C B be the changeofbasis matrix from B to C. Then a. P C B [ x] B = [ x] C for all x in V. b. P C B is the unique matrix P with the property that P [ x] B = [ x] C for all x in V. c. P C B is invertible and (P C B ) 1 = P B C. Linear Transformations A linear transformation from a vector space V to a vector space W is a mapping such that, for all u and v in V and all scalars c, 1. T ( u + v) = T ( u) + T ( v) 2. T (c u) = ct ( u) It is straightforward to show that this definition is equivalent to the requirement that T preserve all linear combinations. That is, is a linear transformation if and only if T (c 1 v c k v k ) = c 1 T ( v 1 ) + + c k T ( v k ) for all v 1,..., v k in V and scalars c 1,..., c k. Properties of Linear Transformations
4 MATH10212 Linear Algebra Brief lecture notes 67 Theorem 6.14 Let be a linear transformation. Then a. T ( 0) = 0) b. T ( v) = T ( v) for all v in V. c. T ( u v) = T ( u) T ( v) for all u and v in V. Theorem 6.15 Let be a linear transformation and let B = { v 1,..., v n } be a spanning set for V. Then T (B) = {T ( v 1 ),..., T ( v n )} spans the range of T. Composition of Linear Transformations and If T : U V S : V W are linear transformations, then the composition of S with T is the mapping S T, defined by (S T )( u) = S(T ( u)) where u is in U. Theorem If T : U V and S : V W are linear transformations, then S T : U W is a linear transformation. Inverses of Linear Transformations A linear transformation is invertible if there is a linear transformation T : W V such that T T = I V and T T = I W In this case, T is called an inverse for T.
5 MATH10212 Linear Algebra Brief lecture notes 68 Theorem 6.17 is unique. If T is an invertible linear transformation, then its inverse The Kernel and Range of a Linear Transformation Let be a linear transformation. The kernel of T, denoted ker(t ), is the set of all vectors in V that are mapped by T to 0 in W. That is, ker (T ) = { v in V : T ( v) = 0} The range of T, denoted range(t ), is the set of all vectors in W that are images of vectors in V under T. That is, range (T ) = {T ( v) : v in V } = { w in W : w = T ( v) for some v in V } Theorem Let be a linear transformation. Then a. The kernel of T is a subspace of V. b. The range of T is a subspace of W. Let be a linear transformation. The rank of T is the dimension of the range of T and is denoted by rank(t ). The nullity of T is the dimension of the kernel of T and is denoted by nullity(t ). Theorem The Rank Theorem Let be a linear transformation from a finitedimensional vector space V into a vector space W. Then rank (T ) + nullity (T ) = dim (T ) OnetoOne and Onto Linear Transformations A linear transformation is called onetoone if T maps distinct vectors in V to distinct vectors in W. If range(t ) = W, then T is called onto.
6 MATH10212 Linear Algebra Brief lecture notes 69 Theorem A linear transformation is onetoone if and only if ker(t ) = { 0}. Theorem Let dimv = dimw = n. Then a linear transformation is onetoone if and only if it is onto. Theorem Let be a onetoone linear transformation. If S = { v 1,..., v k } is a linearly independent set in V, then T (S) = {T ( v 1 ),..., T ( v k )} is a linearly independent set in W. Corollary Let dimv = dimw = n. Then a onetoone linear transformation maps a basis for V to a basis for W. Theorem A linear transformation is invertible if and only if it is onetoone and onto. Isomorphisms of Vector Spaces A linear transformation is called an isomorphism if it is onetoone and onto. If V and W are two vector spaces such that there is an isomorphism from V to W, then we say that V is isomorphic to W and write V W. Theorem 6.25 Let V and W be two finitedimensional vector spaces. Then V is isomorphic to W if and only if dimv = dimw. The Matrix of a Linear Transformation Theorem 6.26 Let V and W be two finitedimensional vector spaces with bases B and C, respectively, where B = { v 1,..., v n }. If is a linear transformation, then the m n matrix A defined by A = [[T ( v 1 )] C [T ( v 2 )] C [T ( v n )] C ]
7 MATH10212 Linear Algebra Brief lecture notes 70 satisfies for every vector v in V. A[ v] B = [T ( v)] C Matrices of Composite and Inverse Linear Transformations Theorem Let U, V and W be finitedimensional vector spaces with bases B, C, and D, respectively. Let T : U V and S : V W be linear transformations. Then [S T ] D B = [S] D C [T ] C B Theorem Let be a linear transformation between n dimensional vector spaces V and W and let B and C be bases for V and W, respectively. Then T is invertible if and only if the matrix [T ] C B is invertible. In this case, ([T ] C B ) 1 = [T 1 ] B C Change of Basis and Similarity Theorem Let V be a finitedimensional vector space with bases B and C and let T : V V be a linear transformation. Then [T ] C = P 1 [T ] B P where P is the changeofbasis matrix from C to B. Let V be a finitedimensional vector space and let T : V V be a linear transformation. Then T is called diagonalizable if there is a basis C for V such that matrix [T ] C is a diagonal matrix. Theorem Version 4 Let A be an n n matrix and let The Fundamental Theorem of Invertible Matrices: be a linear transformation whose matrix [T ] C B with respect to bases B and C of V and W, respectively, is A. The following statements are equivalent:
8 MATH10212 Linear Algebra Brief lecture notes 71 a. A is invertible. b. A x = b has a unique solution for every b in R n. c. A x = 0 has only the trivial solution. d. The reduced row echelon form of A is I n. e. A is a product of elementary matrices. f. rank(a) = n g. nullity(a) = 0 h. The column vectors of A are linearly independent. i. The column vectors of A span R n. j. The column vectors of A form a basis for R n. k. The row vectors of A are linearly independent. l. The row vectors of A span R n. m. The row vectors of A form a basis for R n. n. det A 0 o. 0 is not an eigenvalue of A. p. T is invertible. q. T is onetoone. r. T is onto. s. ker(t ) = { 0} t. range(t ) = W
9 MATH10212 Linear Algebra Brief lecture notes 72 Shortlist of theoretical (bookwork) questions Ideally, you should know the proof of every theorem in the module: there is certainly nothing extra in it, everything belongs to the basics, bare necessities. But to make it easier for you to prepare for the exam, here is a shortlist of theoretical questions, some of which will occur in the exam paper. At the exam, you do not have to reproduce the proofs in the lectures wordbyword. Common sense rules apply: if you are asked to prove something, you cannot just say...because it was proved in the lectures ; on the other hand, there is no need to prove previous lemmas and theorems on which the proof of the required bit was based in the lectures. 1. All the definitions and statements of theorems, lemmas, etc. 2. Explain why it is legitimate to use elementary row operations of the augmented matrix for solving a system of linear equations. 3. Prove that a finite system of vectors is linearly dependent if and only if (at least) one of them is a linear combination of the others. 4. Prove that e.r.o.s do not alter the span of the rows of a matrix. 5. Prove that (AB) T = B T A T. 6. Prove that A + A T and AA T are symmetric matrices. 7. Prove that if A is an invertible matrix, then a linear system A x = b has a unique solution. 8. Prove that (AB) 1 = B 1 A 1 if A 1 and B 1 exist. 9. Prove that if the r.r.e.f. of a square matrix A is I, then A is a product of elementary matrices. 10. Prove that if a square matrix A is a product of elementary matrices, then A is invertible. 11. Prove that a right or left inverse of a square matrix is a genuine twosided inverse of it. 12. Explain why the Gauss Jordan double matrix method works for finding the inverse matrix. 13. Prove that the solution set of any homogeneous system A m n x = 0 is a subspace of R n. 14. Prove the Rank Nullity Theorem for matrices. 15. Suppose that vectors e 1,..., e k form a basis of a vector (sub)space U. Prove that then any vector in U can be uniquely represented as a linear combination of the e i. 16. Prove that every linear transformation is a matrix transformation.
10 MATH10212 Linear Algebra Brief lecture notes Prove that if a matrix is orthogonally diagonalizable, then it is symmetric. 18. Prove that the set E λ of all eigenvectors of an n n matrix A corresponding to an eigenvalue λ together with the zero vector is a subspace of R n. 19. Prove that the product of two orthogonal matrices (of the same size) is also an orthogonal matrix, and that the inverse of an orthogonal matrix is an orthogonal matrix. 20. Prove that eigenvectors for pairwise different eigenvalues are linearly independent. 21. Prove that similar matrices have the same eigenvalues, the same rank, and the same determinant. 22. Prove that det(ab) = det A det B. 23. Prove that the determinant of an orthogonal matrix is 1 or Prove that eigenvectors of a symmetric matrix corresponding to different eigenvalues are orthogonal. 25. Prove that the kernel of a linear transformation is a subspace of V, and the range of T is a subspace of W. 26. Prove that a linear transformation is onetoone if and only if ker(t ) = { 0}.
Subspaces, Basis, Dimension, and Rank
MATH10212 Linear Algebra Brief lecture notes 30 Subspaces, Basis, Dimension, and Rank Definition. A subspace of R n is any collection S of vectors in R n such that 1. The zero vector 0 is in S. 2. If u
More informationMATH 2300 Sample Proofs
MATH 2300 Sample Proofs This document contains a number of theorems, the proofs of which are at a difficulty level where they could be put on a test or exam. This should not be taken as an indication that
More informationAdvanced Linear Algebra Math 4377 / 6308 (Spring 2015) May 12, 2015
Final Exam Advanced Linear Algebra Math 4377 / 638 (Spring 215) May 12, 215 4 points 1. Label the following statements are true or false. (1) If S is a linearly dependent set, then each vector in S is
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 informationMath 4242 Sec 40. Supplemental Notes + Homework 8 (With Solutions)
Math 4242 Sec 40 Supplemental Notes + Homework 8 (With Solutions) 1 Infinite Bases and Dimension Though in this course we care mostly about vector spaces with finite bases, even for this it is convenient
More informationMath 322 Definitions and Theorems
Math 322 Definitions and Theorems Theorem 1.1 (Cancellation Law for Vector Addition). If x,y, and z are vectors in a vector space V such that x+z = y+z, then x = y. Corollary 1. The zero vector in a vector
More informationUnit 17 The Theory of Linear Systems
Unit 17 The Theory of Linear Systems In this section, we look at characteristics of systems of linear equations and also of their solution sets. Theorem 17.1. For any system of linear equations A x = b,
More informationc 1 v 1 + c 2 v c k v k
Definition: A vector space V is a nonempty set of objects, called vectors, on which the operations addition and scalar multiplication have been defined. The operations are subject to ten axioms: For any
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, 2012 CONTENTS LINEAR ALGEBRA AND
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 informationfind the following.
MATHS Linear Algebra, March 6, 6 Time Alloted: minutes Aids permitted: Casio FX99 or Sharp EL5 calculator Solutions to test: Given that the reduced echelon form of 4 4 A = 3 6 3 9 is R = 4 9 3 find the
More information(Practice)Exam in Linear Algebra
(Practice)Exam in Linear Algebra First Year at The Faculties of Engineering and Science and of Health This test has 9 pages and 15 problems. In twosided print. It is allowed to use books, notes, photocopies
More informationSection 4.1: Introduction to Linear Spaces
Section 4.1: Introduction to Linear Spaces Definition: A vector space V is a collection of elements with a rule for addition and scalar multiplication, which is closed under addition and scalar multiplication,
More informationRANK AND NULLITY. x 1. x m
RANK AND NULLITY. The row and column spaces Let A be an m n matrix. Then A has n columns, each of which is a vector in R m. The linear span of the columns is a subspace of R n. It s called the column space
More informationMATH 304 Linear Algebra Lecture 11: Basis and dimension.
MATH 304 Linear Algebra Lecture 11: Basis and dimension. Linear independence Definition. Let V be a vector space. Vectors v 1,v 2,...,v k V are called linearly dependent if they satisfy a relation r 1
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 informationVector Spaces and Linear Transformations
Vector Spaces and Linear Transformations Beifang Chen Fall 6 Vector spaces A vector space is a nonempty set V whose objects are called vectors equipped with two operations called addition and scalar multiplication:
More informationMath 308 Final Exam Winter 2015, Form Bonus. of (10) 135
Math 308 Final Exam Winter 015, 318015 Your Name Your Signature Student ID # Points 1.. 3. 4. 5. 6. 7. 8. 9. 10. 11. Form Bonus of 50 13 1 17 8 3 7 6 3 4 6 6 (10) 135 No books are allowed. But you are
More informationLinear Algebra A Summary
Linear Algebra A Summary Definition: A real vector space is a set V that is provided with an addition and a multiplication such that (a) u V and v V u + v V, (1) u + v = v + u for all u V en v V, (2) u
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 informationMA 52 May 9, Final Review
MA 5 May 9, 6 Final Review This packet contains review problems for the whole course, including all the problems from the previous reviews. We also suggest below problems from the textbook for chapters
More informationMath 4377/6308 Advanced Linear Algebra
Math 4377/6308 Advanced Linear Algebra 2.2 Properties of Linear Transformations, Matrices. Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377 Jiwen
More informationMatrix Inverses. Since the linear system. can be written as. where. ,, and,
Matrix Inverses Consider the ordinary algebraic equation and its solution shown below: Since the linear system can be written as where,, and, (A = coefficient matrix, x = variable vector, b = constant
More informationGRE math study group Linear algebra examples D Joyce, Fall 2011
GRE math study group Linear algebra examples D Joyce, Fall 20 Linear algebra is one of the topics covered by the GRE test in mathematics. Here are the questions relating to linear algebra on the sample
More informationRank and linear transformations
Rank and linear transformations Math 4, Introduction to Linear Algebra Friday, February, Recall from Wednesday... Important characteristic of a basis Theorem. Given a basis B {v,...,v k } of subspace S,
More informationReview: Vector space
Math 2F Linear Algebra Lecture 13 1 Basis and dimensions Slide 1 Review: Subspace of a vector space. (Sec. 4.1) Linear combinations, l.d., l.i. vectors. (Sec. 4.3) Dimension and Base of a vector space.
More informationFurther linear algebra. Chapter III. Revision of linear algebra.
Further linear algebra. Chapter III. Revision of linear algebra. Andrei Yafaev As in the previous chapter, we consider a field k. Typically, k = Q, R, C, F p. A vector space over a field k (one also says
More informationMAT188H1F  Linear Algebra  Fall Solutions to Term Test 2  November 17, 2015
MAT88HF  Linear Algebra  Fall 25 Solutions to Term Test 2  November 7, 25 Time allotted: minutes. Aids permitted: Casio FX99 or Sharp EL52 calculator. General Comments:. The results on this test were
More informationFat, Square and Thin Matrices  Number of Solutions to Systems of Linear Equations
Fat, Square and Thin Matrices  Number of Solutions to Systems of Linear Equations (With Answers to True/False Questions posted by Dr. Webb) March 30, 2016 Introduction The goal of this short article is
More informationUnit 20 Linear Dependence and Independence
Unit 20 Linear Dependence and Independence The idea of dimension is fairly intuitive. Consider any vector in R m, (a 1, a 2, a 3,..., a m ). Each of the m components is independent of the others. That
More informationMath 115AH HW 4 Selected Solutions
Math 115AH HW 4 Selected Solutions Yehonatan Sella 3.1.1 Which of the following functions T from R 2 to R 2 are linear transformations? a) T (x 1, x 2 ) = (1 + x 1, x 2 ). This is not a linear transformation.
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
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 informationSUBSPACES. Chapter Introduction. 3.2 Subspaces of F n
Chapter 3 SUBSPACES 3. Introduction Throughout this chapter, we will be studying F n, the set of all n dimensional column vectors with components from a field F. We continue our study of matrices by considering
More informationM341 (56140), Sample Final Exam Solutions
M4 (5640), Sample Final Exam Solutions Let V be an ndimensional vector space and W be an mdimensional vector space a) Suppose n < m Show that there is no linear transformation L: V W such that L is onto
More informationMore Linear Algebra Study Problems
More Linear Algebra Study Problems The final exam will cover chapters 3 except chapter. About half of the exam will cover the material up to chapter 8 and half will cover the material in chapters 93.
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationJanuary 1984 LINEAR ALGEBRA QUESTIONS FROM THE ADMISSION TO CANDIDACY EXAM
LINEAR ALGEBRA QUESTIONS FROM THE ADMISSION TO CANDIDACY EXAM The following is a more or less complete list of the linear algebra questions that have appeared on the admission to candidacy exam for the
More informationMATH 2030: EIGENVALUES AND EIGENVECTORS
MATH 200: EIGENVALUES AND EIGENVECTORS Eigenvalues and Eigenvectors of n n matrices With the formula for the determinant of a n n matrix, we can extend our discussion on the eigenvalues and eigenvectors
More informationLinear Transformations and the RankNullity Theorem. T (u + v) = T (u) + T (v), T (cu) = ct (u),
Math 4326 Fall 2016 Linear Transformations and the RankNullity Theorem In these notes, I will present everything we know so far about linear transformations. This material comes from sections 1.7, 1.8,
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 informationSOLUTIONS TO PROBLEM SET 6
SOLUTIONS TO PROBLEM SET 6 18.6 SPRING 16 Note the difference of conventions: these solutions adopt that the characteristic polynomial of a matrix A is det A xi while the lectures adopt the convention
More informationMath 24 Winter 2010 Wednesday, February 24
(.) TRUE or FALSE? Math 4 Winter Wednesday, February 4 (a.) Every linear operator on an ndimensional vector space has n distinct eigenvalues. FALSE. There are linear operators with no eigenvalues, and
More informationMath Final Review Dec 10, 2010
Math 301001 Final Review Dec 10, 2010 General Points: Date and time: Monday, December 13, 10:30pm 12:30pm Exam topics: Chapters 1 4, 5.1, 5.2, 6.1, 6.2, 6.4 There is just one fundamental way to prepare
More information2.5 Spaces of a Matrix and Dimension
38 CHAPTER. MORE LINEAR ALGEBRA.5 Spaces of a Matrix and Dimension MATH 94 SPRING 98 PRELIM # 3.5. a) Let C[, ] denote the space of continuous function defined on the interval [, ] (i.e. f(x) is a member
More informationLinear Algebra Prerequisites  continued. Jana Kosecka
Linear Algebra Prerequisites  continued Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html kosecka@cs.gmu.edu Matrices meaning m points from ndimensional space n x m matrix transformation Covariance
More informationSimilarity 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 informationThis MUST hold matrix multiplication satisfies the distributive property.
The columns of AB are combinations of the columns of A. The reason is that each column of AB equals A times the corresponding column of B. But that is a linear combination of the columns of A with coefficients
More informationUsing the three elementary row operations we may rewrite A in an echelon form as
Rank, RowReduced Form, and Solutions to Example 1 Consider the matrix A given by Using the three elementary row operations we may rewrite A in an echelon form as or, continuing with additional row operations,
More information(a) Compute the dimension of the kernel of T and a basis for the kernel. The kernel of T is the nullspace of A, so we row reduce A to find
Scores Name, Section # #2 #3 #4 #5 #6 #7 #8 Midterm 2 Math 27W, Linear Algebra Directions. You have 0 minutes to complete the following 8 problems. A complete answer will always include some kind of work
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 informationDe nitions of Linear Algebra Terms
De nitions of Linear Algebra Terms In order to learn and understand mathematics, it is necessary to understand the meanings of the terms (vocabulary words) that are used This document contains de nitions
More informationMath 110, Spring 2015: Homework 10 Solutions
Math 0, Spring 205: Homework 0 Solutions Section 7 Exercise 73: Let T be a linear operator on a finitedimensional vector space V such that the characteristic polynomial of T splits, and let λ, λ 2,, λ
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 informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Coordinates and linear transformations (Leon 3.5, 4.1 4.3) Coordinates relative to a basis Change of basis, transition matrix Matrix
More informationMath 2040: Matrix Theory and Linear Algebra II Solutions to Assignment 2
ection 4.5 Dimension Math 2040: Matrix Theory and Linear Algebra II olutions to Assignment 2 4.5.6. Problem Restatement: Find (a) a basis, and (b) state the dimension of the subspace a + 6b c { 6a 2b 2c
More informationLecture 19: Section 4.4
Lecture 19: Section 4.4 Shuanglin Shao November 11, 2013 Coordinate System in Linear Algebra. (1). Recall that S = {v 1, v 2,, v r } is linearly independent if the equation c 1 v 1 + + c r v r = 0 implies
More informationB such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix
Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.
More informationOrthogonal Transformations Math 217 Professor Karen Smith
Definition: A linear transformation R n Theorem: If R n Orthogonal Transformations Math 217 Professor Karen Smith (c)2015 UM Math Dept licensed under a Creative Commons ByNCSA 4.0 International License.
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 informationChapters 78: Linear Algebra
Sections 75, 78 & 81 Solutions 1 A linear system of equations of the form 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
More information6 Inner Product Spaces
Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space
More information1 Review. 1.1 Vector Spaces
1 Review 1.1 Vector Spaces Let F be a field. An F vector space or vector space over F is a nonempty set V with elements called vectors and two operations: addition of vectors and multiplication of vectors
More informationSOLUTIONS TO HOMEWORK #7, MATH 54 SECTION 001, SPRING 2012
SOLUTIONS TO HOMEWORK #7, MATH 54 SECTION, SPRING JASON FERGUSON Beware of typos These may not be the only ways to solve these problems In fact, these may not even be the best ways to solve these problems
More informationMath 22 Final Exam 1
Math 22 Final Exam. (36 points) Determine if the following statements are true or false. In each case give either a short justification or example (as appropriate) to justify your conclusion. T F (a) If
More information(u, Av) = (A T u,v), (6.4)
216 SECTION 6.1 CHAPTER 6 6.1 Hermitian Operators HERMITIAN, ORTHOGONAL, AND UNITARY OPERATORS In Chapter 4, we saw advantages in using bases consisting of eigenvectors of linear operators in a number
More information2.6 The Inverse of a Square Matrix
200/2/6 page 62 62 CHAPTER 2 Matrices and Systems of Linear Equations 0 0 2 + i i 2i 5 A = 0 9 0 54 A = i i 4 + i 2 0 60 i + i + 5i 26 The Inverse of a Square Matrix In this section we investigate the
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 informationYORK UNIVERSITY Faculty of Science and Engineering MATH M Test #2 Solutions
YORK UNIVERSITY Faculty of Science and Engineering MATH 2022 3.00 M Test #2 Solutions 1. (10 pts) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this question each
More information1. (16 pts) Find all matrices that commute with A = 0 1. Write B =. Setting AB = BA and equating the four components of the product matrices,
. (6 pts) Find all matrices that commute with A = [ ]. 0 [ ] a b Write B =. Setting AB = BA and equating the four components of the product matrices, c d [ ] a b we see that c = 0 and a = d, so all matrices
More information2: LINEAR TRANSFORMATIONS AND MATRICES
2: LINEAR TRANSFORMATIONS AND MATRICES STEVEN HEILMAN Contents 1. Review 1 2. Linear Transformations 1 3. Null spaces, range, coordinate bases 2 4. Linear Transformations and Bases 4 5. Matrix Representation,
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 informationThere are two sets of tests to see if W is a subspace of V. The First set of tests is: Note: A subspace is also closed under subtraction.
1 Vector Spaces Definition 1.1 (Vector Space). Let V be a set, called the vectors, and F be a field, called the scalars. Let + be a binary operation on V with respect to which V is a commutative Group.
More informationWe have seen that a projection P:V V on a finite vector space can be represented by a matrix of the form
14A. Rank of an Endomorphism 1 Rank of an endomorphism We have seen that a projection P:V V on a finite vector space can be represented by a matrix of the form 1 L 0 O M 1 M = 0 M O 0 L 0 with respect
More informationLinear Algebra and Matrices
LECTURE Linear Algebra and Matrices Before embarking on a study of systems of differential equations we will first review, very quickly, some fundamental objects and operations in linear algebra.. Matrices
More informationMatrix Multiplication Chapter II Matrix Analysis. By Gokturk Poyrazoglu The State University of New York at Buffalo BEST Group Winter Lecture Series
Matrix Multiplication Chapter II Matrix Analysis By Gokturk Poyrazoglu The State University of New York at Buffalo BEST Group Winter Lecture Series Outline 1. Basic Linear Algebra 2. Vector Norms 3. Matrix
More informationTheorem (Schur). Let T be a linear operator on a finitedimensional inner product space V. Suppose that the characteristic polynomial of T
Section 6.4: Lemma 1. Let T be a linear operator on a finitedimensional inner product space V. If T has an eigenvector, then so does T. Proof. Let v be an eigenvector of T corresponding to the eigenvalue
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 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 information2 Review of Linear Algebra and Matrices
5 2 Review of Linear Algebra and Matrices 2.1 Vector Spaces 2.1 Definition: A (real) vector space consists of a non empty set V of elements called vectors, and two operations: (1) Addition is defined for
More informationSymmetric Matrices and Quadratic Forms
7 Symmetric Matrices and Quadratic Forms 7.1 DIAGONALIZAION OF SYMMERIC MARICES SYMMERIC MARIX A symmetric matrix is a matrix A such that. A A Such a matrix is necessarily square. Its main diagonal entries
More information1. Linear systems of equations. Chapters 78: Linear Algebra. Solution(s) of a linear system of equations. Row operations.
A linear system of equations of the form Sections 75 78 & 8 a x + a x + + a n x n = b a x + a x + + a n x n = b a m x + a m x + + a mn x n = b m can be written in matrix form as AX = B where a a a n x
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 informationMath 210 Linear Algebra Fall 2014
Math 210 Linear Algebra Fall 2014 Instructor's Name: Office Location: Office Hours: Office Phone: Email: Course Description This is a first course in vectors, matrices, vector spaces, and linear transformations.
More information1 If T is linear, then T(0 ) = 0. 2 T is linear T(cx + y) = ct(x) + T(y) x, y V, c F. 3 If T is linear, then T(x y) = T(x) T(y) x, y V
Linear Transformations Chapter 2 Linear Transformations and Matrices PerOlof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra We call a
More informationPOL502: Linear Algebra
POL502: Linear Algebra Kosuke Imai Department of Politics, Princeton University December 12, 2005 1 Matrix and System of Linear Equations Definition 1 A m n matrix A is a rectangular array of numbers with
More informationSection 3.6. [tx, ty] = t[x, y] S.
Section 3.6. (a) Let S be the set of vectors [x, y] satisfying x = 2y. Then S is a vector subspace of R 2. For (i) [, ] S as x = 2y holds with x = and y =. (ii) S is closed under addition. For let [x,
More informationMath 480 Diagonalization and the Singular Value Decomposition. These notes cover diagonalization and the Singular Value Decomposition.
Math 480 Diagonalization and the Singular Value Decomposition These notes cover diagonalization and the Singular Value Decomposition. 1. Diagonalization. Recall that a diagonal matrix is a square matrix
More informationMAT 2038 LINEAR ALGEBRA II Instructor: Engin Mermut Course assistant: Zübeyir Türkoğlu web:
MAT 2038 LINEAR ALGEBRA II 15.05.2015 Dokuz Eylül University, Faculty of Science, Department of Mathematics Instructor: Engin Mermut Course assistant: Zübeyir Türkoğlu web: http://kisi.deu.edu.tr/engin.mermut/
More informationLinear Algebra. Vector Spaces. Keith E. Emmert. January 6, Tarleton State University. Linear Algebra. Keith E. Emmert.
Algebra s Tarleton State University January 6, 2012 Spanning a Common Notation Definition N = {1, 2, 3,...} is the set of natural numbers. W = {0, 1, 2, 3,...} is the set of whole numbers. Z = { 3, 2,
More informationMAT 242 Test 2 SOLUTIONS, FORM T
MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these
More information590 Notes July 8, Theorem (1.11). Let W be a subspace of a finitedimensional vector space V. Then
59 Notes July 8, 214 Theorem (111) Let W be a subspace of a finitedimensional vector space V Then 1 W is finite dimensional 2 If dim(w ) = dim(v ), Then W = V Proof of theorem 111 Let dim(v ) = n If W
More informationMatrices, vectors, and vector spaces
Chapter 2 Matrices, vectors, and vector spaces Reading Revision, vectors and matrices, vector spaces, subspaces, linear independence and dependence, bases and dimension, rank of a matrix, linear transformations
More informationLinear Algebra Test File Spring Test #1
Linear Algebra Test File Spring 2015 Test #1 For problems 13, consider the following system of equations. Do not use your calculator. x + y  2z = 0 3x + 2y + 4z = 10 2x + y + 6z = 10 1.) Solve the system
More informationThe matrix equation Ax = b can be written in the equivalent form
Last lecture (revision) Let A = (a ij ) be an n mmatrix and let a i = be column i of A ai a ni The matrix equation Ax = b can be written in the equivalent form x a + x 2 a 2 + + x m a m = b Claim The
More informationMath 333  Practice Exam 2 with Some Solutions
Math 333  Practice Exam 2 with Some Solutions (Note that the exam will NOT be this long) Definitions (0 points) Let T : V W be a transformation Let A be a square matrix (a) Define T is linear (b) Define
More informationγ ), which is equal to rank([t ]γ β β ) by definition. Let T (x) R(T ), for some x V. Let f : R(T ) R(L [T ]
Chapter 3 Defn. (p. 52) If A M m n (F ), we define the rank of A, denoted rank(a), to be the rank of the linear transformation L A : F n F m. Fact 0. Let V be a vector space of dimension n over F with
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationLesson 7 Subspaces; Image and Kernel
Lesson 7 Subspaces; Image and Kernel Math 21b February 21, 2007 Announcements Homework for February 23: 3.1: 10,22,34,44,54,38*,48*,46* My office hours: Monday 24, Tuesday 35 in SC 323 Definition A subset
More information