Chapter 6 Inner product spaces

Size: px
Start display at page:

Download "Chapter 6 Inner product spaces"

Transcription

1 Chapter 6 Inner product spaces

2 6.1 Inner products and norms Definition 1 Let V be a vector space over F. An inner product on V is a function, : V V F such that the following conditions hold. x+z,y = x,y + z,y cx,y = c x,y x,y = x,y x,x > 0 if x 0 and 0,0 = 0.

3 Definition 2 The adjoint of an m n matrix X is the n m matrix A such that (A ) ij = A ji Theorem 1 (6.1) x,y +z = x,y + x,z x,cy = c x,y x,0, = 0,x = 0 x,x = 0 if and only if x = 0 If x,y = x,z for every x V, then y = z.

4 Definition 3 Let V be an inner product space. For x V, the norm of x is x = x,x. Theorem 2 (6.2) Let V be an inner product space over F. For x,y V and c F (a) cx = c x (b) x = 0 if and only if x = 0 and x 0 for any x. (c) (Cauchy-Schwarz Inequality) x, y x y (d) (Triangle Inequality) x + y x + y

5 Proof. (c) Expand x cy 2 and apply with c = x,y y,y (d) Note that x,y + y,x = 2Re x,y and Re x,y x,y. Definition 4 Vectors x,y V are called orthogonal if x,y = 0. A set of vectors S V is called orthogonal if any two distinct vectors are orthogonal.

6 S is called orthonormal if it is orthogonal and x = 1 for every x S. Odd Town Problem There are n people living in an odd town and they form clubs. A club must contain an odd number of members and for any two distinct clubs there must be an even (possibly zero) number of people in both of them. What is the maxiumum number of clubs that can be formed?

7 6.2 The Gram-Schmidt orthogonalization Definition 5 An ordered basis which is orthonormal is called an orthonormal basis. Theorem 3 (6.3) Suppose S = {v 1,...,v k } is an orthogonal subset of V such that v i 0. For y span(s), y = k i=1 y,v i v i 2v i. Corollary 4 Any orthogonal set of non-zero vectors is linearly independent.

8 Theorem 5 (6.4) (Gram-Schmidt algorithm) Let S = {w 1,...,w n } be linearly independent subset. Define S = {v 1,...,v n } as follows, v 1 := w 1 and for k 2 v k = w k k 1 j=1 w k,v j v j 2 v j. Then S is orthogonal and span(s ) = span(s) Proof. This is induction on S. For the inductive step, first check that v k 0 and then compute v k,v i for i < k. dim(span(s k )) = dim(span(s k )) because S k is linearly independent. Theorem 6 (6.5) Let V be a finite-dimensional inner product space and let β = {v 1,...,v n } be an orthonormal basis for V. Then for every x V, x = n i=1 x,v i v i.

9 Corollary 7 Let β = {v 1,...,v n } be orthonormal, let T : V V be linear, and let A = [T] β. Then A ij = T(v j ),v i. Fourier coefficient of x relative to β is x,y where y β. Definition 6 Let S be a non-empty subset of V. Define S = {x V x,y = 0 for every y S}. Note S is a subspace of V.

10 Theorem 8 (6.6) Let W be a finite-dimensional subspace of an inner product space V and let y V. Then there exist unique u W and z W such that y = u+z. Furthermore, if {v 1,...,v k } is an orthonormal basis for W, then u = y,v i v i. Proof. Let u = y,v i v i and z = y u. Check that z W. For the uniqueness u u W and z z W, Corollary 9 For any x W, y x y u and if y x = y u then x = u.

11 Theorem 10 (6.7) Suppose S = {v 1,...,v k } is an orthonormal set in an n-dimensional inner product space V. Then (a) S can be extended to an orthonormal basis {v 1,...,v n } for V; (b) {v k+1,...,v n } is an orthonormal basis for (span(s)). (c) If W is a subspace of V, then dim(w)+dim(w ) = dim(v).

12 6.3 The adjoint of a linear operator Theorem 11 (6.8) Let V be a finite-dimensional inner product space over F, and let g : V F be a linear transformation. Then there exists a unique vector y V such that g(x) = x,y for every x V. Proof. Take an orthonormal basis β = {v,...,v n } and define y = g(v i )v i. Theorem 12 (6.9) Let V be a finite-dimensional inner product space and let T : V V be linear. Then there exists a unique function T : V V such that T(x),y = x,t (y) for all x,y. Furthermore T is linear.

13 Proof. Fix y. Check that g(x) = T(x),y is linear. Theorem 6.8 gives unique y such that g(x) = x,y. Define T (y) = y. Note that T is a function and check that it is linear. Theorem 13 (6.10) Let V be a finite-dimensional inner product space and let β be an orthonormal ordered basis for V. If T is a linear operator on V, then [T ] β = [T] β.

14 Theorem 14 (6.11) Let V be an inner-product space, and let T,U be linear operators on V. Then (a) (T +U) = T +U (b) (ct) = ct (c) (TU) = U T (d) T = T (e) I = I

15 6.4 Normal and self-adjoint operators Lemma 15 Let T be a linear operator on a finite-dimensional inner product space V. If T has an eigenvector, then so does T. Theorem 16 (6.14) (Schur)Let T be a linear operator on a finite-dimensional inner product space V and suppose that the characteristic polynomial of T splits. Then there exists an orthonormal basis β for V such that [T] β is upper triangular. Proof. Induction on n := dim(v). T has a unit eigenvector z; W := span(z). Show that W is T-invariant and dim(w ) = n 1.

16 The characteristic polynomial of T W divided f T (t) (so it splits) and by IH for some γ [T W ] γ is upper triangular. Let β = γ {z}. Then [T] β is upper-triangular. Note:If V has an orthonormal basis of eigenvectors of T, then TT = T T. Definition 7 T : V V is normal if TT = T T. A M n n (F) is normal if AA = A A. Theorem 17 (6.15) Let T be a normal operator on V. Then

17 (a) T(x) = T (x) (b) T ci is normal for every c F. (c) If x is an eigenvector of T, then x is an eigenvector of T. (d) If λ 1,λ 2 are distinct eigenvalues of T with eigenvectors x 1,x 2, then x 1,x 2 = 0. Theorem 18 (6.16) Let T be a linear operator on a finitedimensional complex inner-product space V. Then T is normal if and only if there exists an orthonormal basis for V consisting of eigenvectors of T.

18 Proof. Suppose T is normal. T splits over C and so apply Schur s lemma to get an orthonormal basis β = {v 1,...,v n }. A := [T] β is upper triangular and so T(v 1 ) = A 11 v 1. Show that e k is an eigenvector by induction on k using the fact that A jk = T(v k ),v j. The converse is easy.

19 Definition 8 T : V V is self-adjoint (Hermitian) if T = T. A M n n (F) is Hermitian if A = A. Lemma 19 Let T be a Hermitian operator on a finite-dim inner product space V. Then (a) All eigenvalues of T are real. (b) If V is a real inner product space, then the characteristic polynomial splits. Theorem 20 (6.17) Let T be a linear operator on a finitedimensional real inner-product space V. Then T is Hermitian

20 if and only if there exists an orthonormal basis for V consisting of eigenvectors of T. Proof. The characteristic polynomial splits and so we may apply Schur s lemma. A := [T] β is upper triangular and so A. Thus it must be a diagonal matrix.

21 6.5 Unitary and orthogonal operators and their matrices Definition 9 Let V be a finite-dimensional inner product space over F and let T : V V be linear. If T(x) = x for every x V, then T is called unitary if F = C and orthogonal if F = R. Theorem 21 (6.18) Let T be a linear operator on a finitedimensional inner-product space V. Then the following statements are equivalent. (a) TT = T T = I (b) T(x),T(y) = x,y for all x,y V

22 (c) If β is an orthonormal basis, then so is T(β). (d) There exists an orthonormal basis β such that T(β) is orthonormal. (e) T(x) = x for every x. Proof. (d) (e) Let β = {v 1,...,v n } be orthonormal such that T(β) is orthonormal. Take x V. Then x = a i v i and x 2 = a i 2 = T(x) 2.

23 (e) (a) x,x = x,t T(x) by (e). Thus x,(i T T)(x) = 0 for every x. Set U := I T T. Then U is self-adjoint and so there is an orthonormal basis consisting of eigenvectors of U. Check that U(x) = λx implies λ = 0; U = T 0 ; T T = I; TT = I as well because [T] β is a square matrix. Definition 10 A square matrix A is called orthogonal if A t A = AA t = I and unitary if A A = AA = I. Note: AA = I iff rows of A are orthonormal.

24 A A = I iff columns of A are orthonormal. If λ is an eigenvalue of a unitary (orthogonal) matrix, then λ = 1. Definition 11 A,B M n n (C) (A,B M n n (R)) are unitarily (orthogonally) equivalent if there exists a unitary (orthogonal) matrix P such that A = P 1 BP. Theorem 22 (6.19) Let A M n n (C). Then A is normal if and only if A is unitarily equivalent to a diagonal matrix. Theorem 23 (6.20) Let A M n n (R). Then A is symmetric if and only if A is orthogonally equivalent to a diagonal matrix.

25 Theorem 24 (6.21) (Schur) Let A M n n (F) and suppose f A (t) splits over F. (a) If F = C, then A is unitarily equivalent to a complex uppertriangular matrix. (b) If F = R, then A is orthogonally equivalent to a real uppertriangular matrix.

Math 108b: Notes on the Spectral Theorem

Math 108b: Notes on the Spectral Theorem Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator

More information

Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )

Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v ) Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O

More information

MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.

MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation

More information

Elementary linear algebra

Elementary linear algebra Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The

More information

Math Linear Algebra II. 1. Inner Products and Norms

Math Linear Algebra II. 1. Inner Products and Norms Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,

More information

Chapter 4 Euclid Space

Chapter 4 Euclid Space Chapter 4 Euclid Space Inner Product Spaces Definition.. Let V be a real vector space over IR. A real inner product on V is a real valued function on V V, denoted by (, ), which satisfies () (x, y) = (y,

More information

Linear Algebra. Session 12

Linear Algebra. Session 12 Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)

More information

Schur s Triangularization Theorem. Math 422

Schur s Triangularization Theorem. Math 422 Schur s Triangularization Theorem Math 4 The characteristic polynomial p (t) of a square complex matrix A splits as a product of linear factors of the form (t λ) m Of course, finding these factors is a

More information

Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence)

Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence) Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence) David Glickenstein December 7, 2015 1 Inner product spaces In this chapter, we will only consider the elds R and C. De nition 1 Let V be a vector

More information

Inner products. Theorem (basic properties): Given vectors u, v, w in an inner product space V, and a scalar k, the following properties hold:

Inner products. Theorem (basic properties): Given vectors u, v, w in an inner product space V, and a scalar k, the following properties hold: Inner products Definition: An inner product on a real vector space V is an operation (function) that assigns to each pair of vectors ( u, v) in V a scalar u, v satisfying the following axioms: 1. u, v

More information

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 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 information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors /88 Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix

More information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 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 information

Quantum Computing Lecture 2. Review of Linear Algebra

Quantum Computing Lecture 2. Review of Linear Algebra Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces

More information

Math 24 Spring 2012 Sample Homework Solutions Week 8

Math 24 Spring 2012 Sample Homework Solutions Week 8 Math 4 Spring Sample Homework Solutions Week 8 Section 5. (.) Test A M (R) for diagonalizability, and if possible find an invertible matrix Q and a diagonal matrix D such that Q AQ = D. ( ) 4 (c) A =.

More information

DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.

DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra. DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1

More information

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v

More information

1. General Vector Spaces

1. General Vector Spaces 1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

More information

I. Multiple Choice Questions (Answer any eight)

I. Multiple Choice Questions (Answer any eight) Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY

More information

x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7

x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)

More information

The following definition is fundamental.

The following definition is fundamental. 1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic

More information

Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.

Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces. Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,

More information

Mathematics Department Stanford University Math 61CM/DM Inner products

Mathematics Department Stanford University Math 61CM/DM Inner products Mathematics Department Stanford University Math 61CM/DM Inner products Recall the definition of an inner product space; see Appendix A.8 of the textbook. Definition 1 An inner product space V is a vector

More information

1 Last time: least-squares problems

1 Last time: least-squares problems MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that

More information

Lecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why?

Lecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why? KTH ROYAL INSTITUTE OF TECHNOLOGY Norms for vectors and matrices Why? Lecture 5 Ch. 5, Norms for vectors and matrices Emil Björnson/Magnus Jansson/Mats Bengtsson April 27, 2016 Problem: Measure size of

More information

Diagonalizing Matrices

Diagonalizing Matrices Diagonalizing Matrices Massoud Malek A A Let A = A k be an n n non-singular matrix and let B = A = [B, B,, B k,, B n ] Then A n A B = A A 0 0 A k [B, B,, B k,, B n ] = 0 0 = I n 0 A n Notice that A i B

More information

Linear Algebra Massoud Malek

Linear Algebra Massoud Malek CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

More information

Linear Algebra. Workbook

Linear Algebra. Workbook Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx

More information

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR 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 information

MATH 115A: SAMPLE FINAL SOLUTIONS

MATH 115A: SAMPLE FINAL SOLUTIONS MATH A: SAMPLE FINAL SOLUTIONS JOE HUGHES. Let V be the set of all functions f : R R such that f( x) = f(x) for all x R. Show that V is a vector space over R under the usual addition and scalar multiplication

More information

MATRICES ARE SIMILAR TO TRIANGULAR MATRICES

MATRICES ARE SIMILAR TO TRIANGULAR MATRICES MATRICES ARE SIMILAR TO TRIANGULAR MATRICES 1 Complex matrices Recall that the complex numbers are given by a + ib where a and b are real and i is the imaginary unity, ie, i 2 = 1 In what we describe below,

More information

Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math 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 information

Linear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall A: Inner products

Linear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall A: Inner products Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 6A: Inner products In this chapter, the field F = R or C. We regard F equipped with a conjugation χ : F F. If F =

More information

Math 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 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 information

MATH 235: Inner Product Spaces, Assignment 7

MATH 235: Inner Product Spaces, Assignment 7 MATH 235: Inner Product Spaces, Assignment 7 Hand in questions 3,4,5,6,9, by 9:3 am on Wednesday March 26, 28. Contents Orthogonal Basis for Inner Product Space 2 2 Inner-Product Function Space 2 3 Weighted

More information

Review problems for MA 54, Fall 2004.

Review 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 information

Linear Analysis Lecture 5

Linear Analysis Lecture 5 Linear Analysis Lecture 5 Inner Products and V Let dim V < with inner product,. Choose a basis B and let v, w V have coordinates in F n given by x 1. x n and y 1. y n, respectively. Let A F n n be the

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

More information

Linear algebra 2. Yoav Zemel. March 1, 2012

Linear algebra 2. Yoav Zemel. March 1, 2012 Linear algebra 2 Yoav Zemel March 1, 2012 These notes were written by Yoav Zemel. The lecturer, Shmuel Berger, should not be held responsible for any mistake. Any comments are welcome at zamsh7@gmail.com.

More information

6 Inner Product Spaces

6 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 information

j=1 x j p, if 1 p <, x i ξ : x i < ξ} 0 as p.

j=1 x j p, if 1 p <, x i ξ : x i < ξ} 0 as p. LINEAR ALGEBRA Fall 203 The final exam Almost all of the problems solved Exercise Let (V, ) be a normed vector space. Prove x y x y for all x, y V. Everybody knows how to do this! Exercise 2 If V is a

More information

Lecture 7: Positive Semidefinite Matrices

Lecture 7: Positive Semidefinite Matrices Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.

More information

5 Compact linear operators

5 Compact linear operators 5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.

More information

Linear Algebra Lecture Notes-II

Linear Algebra Lecture Notes-II Linear Algebra Lecture Notes-II Vikas Bist Department of Mathematics Panjab University, Chandigarh-64 email: bistvikas@gmail.com Last revised on March 5, 8 This text is based on the lectures delivered

More information

Linear Algebra Highlights

Linear 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 information

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) ( ) Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

More information

Vector spaces. DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis.

Vector spaces. DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis. Vector spaces DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_fall17/index.html Carlos Fernandez-Granda Vector space Consists of: A set V A scalar

More information

Linear Algebra in Actuarial Science: Slides to the lecture

Linear Algebra in Actuarial Science: Slides to the lecture Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations

More information

Lecture # 3 Orthogonal Matrices and Matrix Norms. We repeat the definition an orthogonal set and orthornormal set.

Lecture # 3 Orthogonal Matrices and Matrix Norms. We repeat the definition an orthogonal set and orthornormal set. Lecture # 3 Orthogonal Matrices and Matrix Norms We repeat the definition an orthogonal set and orthornormal set. Definition A set of k vectors {u, u 2,..., u k }, where each u i R n, is said to be an

More information

SYLLABUS. 1 Linear maps and matrices

SYLLABUS. 1 Linear maps and matrices Dr. K. Bellová Mathematics 2 (10-PHY-BIPMA2) SYLLABUS 1 Linear maps and matrices Operations with linear maps. Prop 1.1.1: 1) sum, scalar multiple, composition of linear maps are linear maps; 2) L(U, V

More information

Best approximation in the 2-norm

Best approximation in the 2-norm Best approximation in the 2-norm Department of Mathematical Sciences, NTNU september 26th 2012 Vector space A real vector space V is a set with a 0 element and three operations: Addition: x, y V then x

More information

Review of some mathematical tools

Review of some mathematical tools MATHEMATICAL FOUNDATIONS OF SIGNAL PROCESSING Fall 2016 Benjamín Béjar Haro, Mihailo Kolundžija, Reza Parhizkar, Adam Scholefield Teaching assistants: Golnoosh Elhami, Hanjie Pan Review of some mathematical

More information

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

More information

Fourier and Wavelet Signal Processing

Fourier and Wavelet Signal Processing Ecole Polytechnique Federale de Lausanne (EPFL) Audio-Visual Communications Laboratory (LCAV) Fourier and Wavelet Signal Processing Martin Vetterli Amina Chebira, Ali Hormati Spring 2011 2/25/2011 1 Outline

More information

Throughout these notes we assume V, W are finite dimensional inner product spaces over C.

Throughout these notes we assume V, W are finite dimensional inner product spaces over C. Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal

More information

Exercise Sheet 1.

Exercise Sheet 1. Exercise Sheet 1 You can download my lecture and exercise sheets at the address http://sami.hust.edu.vn/giang-vien/?name=huynt 1) Let A, B be sets. What does the statement "A is not a subset of B " mean?

More information

Lecture notes: Applied linear algebra Part 1. Version 2

Lecture notes: Applied linear algebra Part 1. Version 2 Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and

More information

1 Linear Algebra Problems

1 Linear Algebra Problems Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and

More information

Real symmetric matrices/1. 1 Eigenvalues and eigenvectors

Real symmetric matrices/1. 1 Eigenvalues and eigenvectors Real symmetric matrices 1 Eigenvalues and eigenvectors We use the convention that vectors are row vectors and matrices act on the right. Let A be a square matrix with entries in a field F; suppose that

More information

7 Bilinear forms and inner products

7 Bilinear forms and inner products 7 Bilinear forms and inner products Definition 7.1 A bilinear form θ on a vector space V over a field F is a function θ : V V F such that θ(λu+µv,w) = λθ(u,w)+µθ(v,w) θ(u,λv +µw) = λθ(u,v)+µθ(u,w) for

More information

Matrix Theory. A.Holst, V.Ufnarovski

Matrix Theory. A.Holst, V.Ufnarovski Matrix Theory AHolst, VUfnarovski 55 HINTS AND ANSWERS 9 55 Hints and answers There are two different approaches In the first one write A as a block of rows and note that in B = E ij A all rows different

More information

GQE ALGEBRA PROBLEMS

GQE ALGEBRA PROBLEMS GQE ALGEBRA PROBLEMS JAKOB STREIPEL Contents. Eigenthings 2. Norms, Inner Products, Orthogonality, and Such 6 3. Determinants, Inverses, and Linear (In)dependence 4. (Invariant) Subspaces 3 Throughout

More information

Vectors. Vectors and the scalar multiplication and vector addition operations:

Vectors. Vectors and the scalar multiplication and vector addition operations: Vectors Vectors and the scalar multiplication and vector addition operations: x 1 x 1 y 1 2x 1 + 3y 1 x x n 1 = 2 x R n, 2 2 y + 3 2 2x = 2 + 3y 2............ x n x n y n 2x n + 3y n I ll use the two terms

More information

MA 265 FINAL EXAM Fall 2012

MA 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 information

Final A. Problem Points Score Total 100. Math115A Nadja Hempel 03/23/2017

Final A. Problem Points Score Total 100. Math115A Nadja Hempel 03/23/2017 Final A Math115A Nadja Hempel 03/23/2017 nadja@math.ucla.edu Name: UID: Problem Points Score 1 10 2 20 3 5 4 5 5 9 6 5 7 7 8 13 9 16 10 10 Total 100 1 2 Exercise 1. (10pt) Let T : V V be a linear transformation.

More information

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 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 information

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA 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 information

A linear algebra proof of the fundamental theorem of algebra

A linear algebra proof of the fundamental theorem of algebra A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional

More information

The Spectral Theorem for normal linear maps

The Spectral Theorem for normal linear maps MAT067 University of California, Davis Winter 2007 The Spectral Theorem for normal linear maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 14, 2007) In this section we come back to the question

More information

A linear algebra proof of the fundamental theorem of algebra

A linear algebra proof of the fundamental theorem of algebra A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional

More information

(v, w) = arccos( < v, w >

(v, w) = arccos( < v, w > MA322 Sathaye Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: For all v

More information

Math 4153 Exam 3 Review. The syllabus for Exam 3 is Chapter 6 (pages ), Chapter 7 through page 137, and Chapter 8 through page 182 in Axler.

Math 4153 Exam 3 Review. The syllabus for Exam 3 is Chapter 6 (pages ), Chapter 7 through page 137, and Chapter 8 through page 182 in Axler. Math 453 Exam 3 Review The syllabus for Exam 3 is Chapter 6 (pages -2), Chapter 7 through page 37, and Chapter 8 through page 82 in Axler.. You should be sure to know precise definition of the terms we

More information

Spectral Theorem for Self-adjoint Linear Operators

Spectral Theorem for Self-adjoint Linear Operators Notes for the undergraduate lecture by David Adams. (These are the notes I would write if I was teaching a course on this topic. I have included more material than I will cover in the 45 minute lecture;

More information

5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.

5.) 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 information

REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable

More information

Math 102 Final Exam - Dec 14 - PCYNH pm Fall Name Student No. Section A0

Math 102 Final Exam - Dec 14 - PCYNH pm Fall Name Student No. Section A0 Math 12 Final Exam - Dec 14 - PCYNH 122-6pm Fall 212 Name Student No. Section A No aids allowed. Answer all questions on test paper. Total Marks: 4 8 questions (plus a 9th bonus question), 5 points per

More information

Symmetric and self-adjoint matrices

Symmetric and self-adjoint matrices Symmetric and self-adjoint matrices A matrix A in M n (F) is called symmetric if A T = A, ie A ij = A ji for each i, j; and self-adjoint if A = A, ie A ij = A ji or each i, j Note for A in M n (R) that

More information

October 25, 2013 INNER PRODUCT SPACES

October 25, 2013 INNER PRODUCT SPACES October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal

More information

1 Inner Product and Orthogonality

1 Inner Product and Orthogonality CSCI 4/Fall 6/Vora/GWU/Orthogonality and Norms Inner Product and Orthogonality Definition : The inner product of two vectors x and y, x x x =.., y =. x n y y... y n is denoted x, y : Note that n x, y =

More information

Numerical Linear Algebra

Numerical Linear Algebra University of Alabama at Birmingham Department of Mathematics Numerical Linear Algebra Lecture Notes for MA 660 (1997 2014) Dr Nikolai Chernov April 2014 Chapter 0 Review of Linear Algebra 0.1 Matrices

More information

MATH 304 Linear Algebra Lecture 34: Review for Test 2.

MATH 304 Linear Algebra Lecture 34: Review for Test 2. MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1

More information

Chapter 1 Vector Spaces

Chapter 1 Vector Spaces Chapter 1 Vector Spaces Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field

More information

MTH 2032 SemesterII

MTH 2032 SemesterII MTH 202 SemesterII 2010-11 Linear Algebra Worked Examples Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2011 ii Contents Table of Contents

More information

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2 Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition

More information

Projection Theorem 1

Projection Theorem 1 Projection Theorem 1 Cauchy-Schwarz Inequality Lemma. (Cauchy-Schwarz Inequality) For all x, y in an inner product space, [ xy, ] x y. Equality holds if and only if x y or y θ. Proof. If y θ, the inequality

More information

Ordinary Differential Equations II

Ordinary Differential Equations II Ordinary Differential Equations II February 23 2017 Separation of variables Wave eq. (PDE) 2 u t (t, x) = 2 u 2 c2 (t, x), x2 c > 0 constant. Describes small vibrations in a homogeneous string. u(t, x)

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW JC Stuff you should know for the exam. 1. Basics on vector spaces (1) F n is the set of all n-tuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication

More information

INNER PRODUCT SPACE. Definition 1

INNER PRODUCT SPACE. Definition 1 INNER PRODUCT SPACE Definition 1 Suppose u, v and w are all vectors in vector space V and c is any scalar. An inner product space on the vectors space V is a function that associates with each pair of

More information

Introduction to Linear Algebra, Second Edition, Serge Lange

Introduction to Linear Algebra, Second Edition, Serge Lange Introduction to Linear Algebra, Second Edition, Serge Lange Chapter I: Vectors R n defined. Addition and scalar multiplication in R n. Two geometric interpretations for a vector: point and displacement.

More information

Linear Algebra Practice Problems

Linear Algebra Practice Problems Linear Algebra Practice Problems Page of 7 Linear Algebra Practice Problems These problems cover Chapters 4, 5, 6, and 7 of Elementary Linear Algebra, 6th ed, by Ron Larson and David Falvo (ISBN-3 = 978--68-78376-2,

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors

More information

NORMS ON SPACE OF MATRICES

NORMS ON SPACE OF MATRICES NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system

More information

MATH 167: APPLIED LINEAR ALGEBRA Least-Squares

MATH 167: APPLIED LINEAR ALGEBRA Least-Squares MATH 167: APPLIED LINEAR ALGEBRA Least-Squares October 30, 2014 Least Squares We do a series of experiments, collecting data. We wish to see patterns!! We expect the output b to be a linear function of

More information

EXAM. Exam 1. Math 5316, Fall December 2, 2012

EXAM. Exam 1. Math 5316, Fall December 2, 2012 EXAM Exam Math 536, Fall 22 December 2, 22 Write all of your answers on separate sheets of paper. You can keep the exam questions. This is a takehome exam, to be worked individually. You can use your notes.

More information

Then x 1,..., x n is a basis as desired. Indeed, it suffices to verify that it spans V, since n = dim(v ). We may write any v V as r

Then x 1,..., x n is a basis as desired. Indeed, it suffices to verify that it spans V, since n = dim(v ). We may write any v V as r Practice final solutions. I did not include definitions which you can find in Axler or in the course notes. These solutions are on the terse side, but would be acceptable in the final. However, if you

More information

PRACTICE PROBLEMS FOR THE FINAL

PRACTICE PROBLEMS FOR THE FINAL PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show

More information

Math 407: Linear Optimization

Math 407: Linear Optimization Math 407: Linear Optimization Lecture 16: The Linear Least Squares Problem II Math Dept, University of Washington February 28, 2018 Lecture 16: The Linear Least Squares Problem II (Math Dept, University

More information

Economics 204 Summer/Fall 2010 Lecture 10 Friday August 6, 2010

Economics 204 Summer/Fall 2010 Lecture 10 Friday August 6, 2010 Economics 204 Summer/Fall 2010 Lecture 10 Friday August 6, 2010 Diagonalization of Symmetric Real Matrices (from Handout Definition 1 Let δ ij = { 1 if i = j 0 if i j A basis V = {v 1,..., v n } of R n

More information

Linear Algebra 2 Spectral Notes

Linear Algebra 2 Spectral Notes Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex

More information

Lecture 1: Review of linear algebra

Lecture 1: Review of linear algebra Lecture 1: Review of linear algebra Linear functions and linearization Inverse matrix, least-squares and least-norm solutions Subspaces, basis, and dimension Change of basis and similarity transformations

More information