Mathematical foundations  linear algebra


 Brett Alexander
 8 months ago
 Views:
Transcription
1 Mathematical foundations  linear algebra Andrea Passerini Machine Learning
2 Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar multiplication are defined and satisfy for all x, y, x X and λ, µ IR: Addition: associative x + (y + z) = (x + y) + z commutative x + y = y + x identity element 0 X : x + 0 = x inverse element x X x X : x + x = 0 Scalar multiplication: distributive over elements λ(x + y) = λx + λy distributive over scalars (λ + µ)x = λx + µx associative over scalars λ(µx) = (λµ)x identity element 1 IR : 1x = x
3 Properties and operations in vector spaces subspace Any nonempty subset of X being itself a vector space (E.g. projection) linear combination given λ i IR, x i X n λ i x i i=1 span The span of vectors x 1,..., x n is defined as the set of their linear combinations { n } λ i x i, λ i IR i=1
4 Basis in vector space Linear independency A set of vectors x i is linearly independent if none of them can be written as a linear combination of the others Basis A set of vectors x i is a basis for X if any element in X can be uniquely written as a linear combination of vectors x i. Necessary condition is that vectors x i are linearly independent All bases of X have the same number of elements, called the dimension of the vector space.
5 Linear maps Definition Given two vector spaces X, Z, a function f : X Z is a linear map if for all x, y X, λ IR: f (x + y) = f (x) + f (y) f (λx) = λf (x)
6 Linear maps as matrices A linear map between two finitedimensional spaces X, Z of dimensions n, m can always be written as a matrix: Let {x 1,..., x n } and {z 1,..., z m } be some bases for X and Z respectively. For any x X we have: n n f (x) = f ( λ i x i ) = λ i f (x i ) f (x i ) = f (x) = i=1 m a ij z j j=1 i=1 n m m n λ i a ij z j = ( λ i a ij )z j = i=1 j=1 j=1 i=1 m µ j z j j=1
7 Linear maps as matrices Matrix of basis transformation a a 1n M IR m n =... a m1... a mn Mapping from basis coefficients to basis coefficients Mλ = µ
8 Matrix properties transpose Matrix obtained exchanging rows with columns (indicated with M T ). Properties: (MN) T = N T M T trace Sum of diagonal elements of a matrix tr(m) = n i=1 M ii inverse The matrix which multiplied with the original matrix gives the identity MM 1 = I rank The rank of an n m matrix is the dimension of the space spanned by its columns
9 Metric structure Norm A function : X IR + 0 is a norm if for all x, y X, λ IR: x + y x + y λx = λ x x > 0 if x 0 Metric A norm defines a metric d : X X IR + 0 : d(x, y) = x y Note The concept of norm is stronger than that of metric: not any metric gives rise to a norm
10 Dot product Bilinear form A function Q : X X IR is a bilinear form if for all x, y, z X, λ, µ IR: Q(λx + µy, z) = λq(x, z) + µq(y, z) Q(x, λy + µz) = λq(x, y) + µq(x, z) A bilinear form is symmetric if for all x, y X : Q(x, y) = Q(y, x)
11 Dot product Dot product A dot product, : X X IR is a symmetric bilinear form which is positive definite: x, x 0 x X A strictly positive definite dot product satisfies x, x = 0 iff x = 0 Norm Any dot product defines a corresponding norm via: x = x, x
12 Properties of dot product angle The angle θ between two vectors is defined as: cosθ = orthogonal Two vectors are orthogonal if x, z x z x, y = 0 orthonormal A set of vectors {x 1,..., x n } is orthonormal if x i, x j = δ ij where δ ij = 1 if i = j, 0 otherwise.
13 Eigenvalues and eigenvectors Definition Given a matrix M, the real value λ and (nonzero) vector x are an eigenvalue and corresponding eigenvector of M if Spectral decomposition Mx = λx Given an n n symmetric matrix M, it is possible to find an orthonormal set of n eigenvectors. The matrix V having eigenvectors as columns is unitary, that is V T = V 1 M can be factorized as: M = V ΛV T where Λ is the diagonal matrix of eigenvalues corresponding to eigenvectors in V
14 Eigenvalues and eigenvectors Singular matrices A matrix is singular if it has a zero eigenvalue Mx = 0x = 0 A singular matrix has linearly dependent columns: [ ] M1... M n 1 M n x 1. x n 1 = 0 x n
15 Eigenvalues and eigenvectors Singular matrices A matrix is singular if it has a zero eigenvalue Mx = 0x = 0 A singular matrix has linearly dependent columns: M 1 x M n 1 x n 1 + M n x n = 0
16 Eigenvalues and eigenvectors Singular matrices A matrix is singular if it has a zero eigenvalue Mx = 0x = 0 A singular matrix has linearly dependent columns: M n = M 1 x 1 x n + + M n 1 x n 1 x n
17 Eigenvalues and eigenvectors Singular matrices A matrix is singular if it has a zero eigenvalue Mx = 0x = 0 A singular matrix has linearly dependent columns: M n = M 1 x 1 x n + + M n 1 x n 1 x n Determinant The determinant M of a n n matrix M is the product of its eigenvalues A matrix is invertible if its determinant is not zero (i.e. it is not singular)
18 Eigenvalues and eigenvectors Symmetric matrices Eigenvectors corresponding to distinct eigenvalues are orthogonal: λ x, z = Ax, z = (Ax) T z = x T A T z = x T Az = x, Az = µ x, z Note An n n symmetric matrix can have at most n distinct eigenvalues.
19 Positive semidefinite matrix Definition An n n symmetrix matrix M is positive semidefinite if all its eigenvalues are nonnegative. Alternative sufficient and necessary conditions for all x IR n x T Mx 0 there exists a real matrix B s.t. M = B T B
20 Functional analysis Cauchy sequence A sequence (x i ) i IN in a normed space X is called a Cauchy sequence if for all ɛ > 0 there exists n IN s.t. n, n > n x n x n < ɛ. A Cauchy sequence converges to a point x X if Banach and Hilbert spaces lim x n x 0 n A space is complete if all Cauchy sequences in the space converge (to a point in the space). A complete normed space is called a Banach space A complete dot product space is called a Hilbert space
21 Hilbert spaces A Hilbert space can be (and often is) infinite dimensional Infinite dimensional Hilbert spaces are usually required to be separable, i.e. there exists a countable dense subset: for any ɛ > 0 there exists a sequence (x i ) i IN of elements of the space s.t. for all x X min x i x < ɛ x i
Mathematical foundations  linear algebra
Mathematical foundations  linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationLinear 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 informationFunctional Analysis Review
Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all
More informationBasic Calculus Review
Basic Calculus Review Lorenzo Rosasco ISML Mod. 2  Machine Learning Vector Spaces Functionals and Operators (Matrices) Vector Space A vector space is a set V with binary operations +: V V V and : R V
More informationThe 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 informationLinear 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 informationLinear Algebra Review
January 29, 2013 Table of contents Metrics Metric Given a space X, then d : X X R + 0 and z in X if: d(x, y) = 0 is equivalent to x = y d(x, y) = d(y, x) d(x, y) d(x, z) + d(z, y) is a metric is for all
More information5 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 finitedimensional space, there exists a basis consisting of eigenvectors.
More informationA Review of Linear Algebra
A Review of Linear Algebra Mohammad Emtiyaz Khan CS,UBC A Review of Linear Algebra p.1/13 Basics Column vector x R n, Row vector x T, Matrix A R m n. Matrix Multiplication, (m n)(n k) m k, AB BA. Transpose
More informationNORMS 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 informationLinear 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 (ISBN3 = 97868783762,
More informationFall TMA4145 Linear Methods. Exercise set Given the matrix 1 2
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an
More informationLecture 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 informationExample Linear Algebra Competency Test
Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More information2. Review of Linear Algebra
2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear
More informationj=1 u 1jv 1j. 1/ 2 Lemma 1. An orthogonal set of vectors must be linearly independent.
Lecture Notes: Orthogonal and Symmetric Matrices Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Orthogonal Matrix Definition. Let u = [u
More informationReview of Some Concepts from Linear Algebra: Part 2
Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set
More informationLecture II: Linear Algebra Revisited
Lecture II: Linear Algebra Revisited Overview Vector spaces, Hilbert & Banach Spaces, etrics & Norms atrices, Eigenvalues, Orthogonal Transformations, Singular Values Operators, Operator Norms, Function
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Numerical Linear Algebra Background ChoJui Hsieh UC Davis May 15, 2018 Linear Algebra Background Vectors A vector has a direction and a magnitude
More information1. Foundations of Numerics from Advanced Mathematics. Linear Algebra
Foundations of Numerics from Advanced Mathematics Linear Algebra Linear Algebra, October 23, 22 Linear Algebra Mathematical Structures a mathematical structure consists of one or several sets and one or
More informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationMath Solutions to homework 5
Math 75  Solutions to homework 5 Cédric De Groote November 9, 207 Problem (7. in the book): Let {e n } be a complete orthonormal sequence in a Hilbert space H and let λ n C for n N. Show that there is
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationMath 224, Fall 2007 Exam 3 Thursday, December 6, 2007
Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007 You have 1 hour and 20 minutes. No notes, books, or other references. You are permitted to use Maple during this exam, but you must start with a blank
More informationDefinitions for Quizzes
Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does
More informationQuantum 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 informationCS 143 Linear Algebra Review
CS 143 Linear Algebra Review Stefan Roth September 29, 2003 Introductory Remarks This review does not aim at mathematical rigor very much, but instead at ease of understanding and conciseness. Please see
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationReview 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 information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 201617/08aops
More informationSelçuk Demir WS 2017 Functional Analysis Homework Sheet
Selçuk Demir WS 2017 Functional Analysis Homework Sheet 1. Let M be a metric space. If A M is nonempty, we say that A is bounded iff diam(a) = sup{d(x, y) : x.y A} exists. Show that A is bounded iff there
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationLecture 3: Review of Linear Algebra
ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters, transforms,
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationIr 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 informationLecture 3: Review of Linear Algebra
ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,
More informationKnowledge 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 informationOptimization Theory. A Concise Introduction. Jiongmin Yong
October 11, 017 16:5 wsbook9x6 Book Title Optimization Theory 01708Lecture Notes page 1 1 Optimization Theory A Concise Introduction Jiongmin Yong Optimization Theory 01708Lecture Notes page Optimization
More informationHilbert Spaces. Contents
Hilbert Spaces Contents 1 Introducing Hilbert Spaces 1 1.1 Basic definitions........................... 1 1.2 Results about norms and inner products.............. 3 1.3 Banach and Hilbert spaces......................
More informationVectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1. x 2. x =
Linear Algebra Review Vectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1 x x = 2. x n Vectors of up to three dimensions are easy to diagram.
More informationELE/MCE 503 Linear Algebra Facts Fall 2018
ELE/MCE 503 Linear Algebra Facts Fall 2018 Fact N.1 A set of vectors is linearly independent if and only if none of the vectors in the set can be written as a linear combination of the others. Fact N.2
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 5: Numerical Linear Algebra ChoJui Hsieh UC Davis April 20, 2017 Linear Algebra Background Vectors A vector has a direction and a magnitude
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A pdimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p
More informationSPECTRAL THEOREM FOR COMPACT SELFADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELFADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Selfadjoint
More informationLecture 3: Linear Algebra Review, Part II
Lecture 3: Linear Algebra Review, Part II Brian Borchers January 4, Linear Independence Definition The vectors v, v,..., v n are linearly independent if the system of equations c v + c v +...+ c n v n
More informationProblem Set 1. Homeworks will graded based on content and clarity. Please show your work clearly for full credit.
CSE 151: Introduction to Machine Learning Winter 2017 Problem Set 1 Instructor: Kamalika Chaudhuri Due on: Jan 28 Instructions This is a 40 point homework Homeworks will graded based on content and clarity
More informationMatrices and Vectors. Definition of Matrix. An MxN matrix A is a twodimensional array of numbers A =
30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a twodimensional array of numbers A = a 11 a 12... a 1N a 21 a 22... a 2N...... a M1 a M2... a MN A matrix can
More information1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?
. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationProperties of Matrices and Operations on Matrices
Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,
More informationLinear Algebra  Part II
Linear Algebra  Part II Projection, Eigendecomposition, SVD (Adapted from Sargur Srihari s slides) Brief Review from Part 1 Symmetric Matrix: A = A T Orthogonal Matrix: A T A = AA T = I and A 1 = A T
More informationElementary 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(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 informationMath 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 informationMath Bootcamp An pdimensional vector is p numbers put together. Written as. x 1 x =. x p
Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An pdimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
More informationDuke University, Department of Electrical and Computer Engineering Optimization for Scientists and Engineers c Alex Bronstein, 2014
Duke University, Department of Electrical and Computer Engineering Optimization for Scientists and Engineers c Alex Bronstein, 2014 Linear Algebra A Brief Reminder Purpose. The purpose of this document
More informationLECTURE 7. k=1 (, v k)u k. Moreover r
LECTURE 7 Finite rank operators Definition. T is said to be of rank r (r < ) if dim T(H) = r. The class of operators of rank r is denoted by K r and K := r K r. Theorem 1. T K r iff T K r. Proof. Let T
More informationChapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors
Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there
More informationEcon Slides from Lecture 7
Econ 205 Sobel Econ 205  Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for
More informationSPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT
SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT Abstract. These are the letcure notes prepared for the workshop on Functional Analysis and Operator Algebras to be held at NITKarnataka,
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationMATH 304 Linear Algebra Lecture 20: The GramSchmidt process (continued). Eigenvalues and eigenvectors.
MATH 304 Linear Algebra Lecture 20: The GramSchmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v
More informationMatrix Algebra: Summary
May, 27 Appendix E Matrix Algebra: Summary ontents E. Vectors and Matrtices.......................... 2 E.. Notation.................................. 2 E..2 Special Types of Vectors.........................
More informationDefinition 1. A set V is a vector space over the scalar field F {R, C} iff. there are two operations defined on V, called vector addition
6 Vector Spaces with Inned Product Basis and Dimension Section Objective(s): Vector Spaces and Subspaces Linear (In)dependence Basis and Dimension Inner Product 6 Vector Spaces and Subspaces Definition
More informationI. 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 : Sep24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY
More informationTutorials in Optimization. Richard Socher
Tutorials in Optimization Richard Socher July 20, 2008 CONTENTS 1 Contents 1 Linear Algebra: Bilinear Form  A Simple Optimization Problem 2 1.1 Definitions........................................ 2 1.2
More informationMATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian.
MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. Spanning set Let S be a subset of a vector space V. Definition. The span of the set S is the smallest subspace W V that contains S. If
More information7 : APPENDIX. Vectors and Matrices
7 : APPENDIX Vectors and Matrices An ntuple vector x is defined as an ordered set of n numbers. Usually we write these numbers x 1,...,x n in a column in the order indicated by their subscripts. The transpose
More informationAssignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.
Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has
More informationLarge Scale Data Analysis Using Deep Learning
Large Scale Data Analysis Using Deep Learning Linear Algebra U Kang Seoul National University U Kang 1 In This Lecture Overview of linear algebra (but, not a comprehensive survey) Focused on the subset
More informationLecture 2: Linear Algebra
Lecture 2: Linear Algebra Rajat Mittal IIT Kanpur We will start with the basics of linear algebra that will be needed throughout this course That means, we will learn about vector spaces, linear independence,
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 WeiTa Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationNOTES (1) FOR MATH 375, FALL 2012
NOTES 1) FOR MATH 375, FALL 2012 1 Vector Spaces 11 Axioms Linear algebra grows out of the problem of solving simultaneous systems of linear equations such as 3x + 2y = 5, 111) x 3y = 9, or 2x + 3y z =
More informationLinear 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 informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For
More informationHilbert Spaces: InfiniteDimensional Vector Spaces
Hilbert Spaces: InfiniteDimensional Vector Spaces PHYS 500  Southern Illinois University October 27, 2016 PHYS 500  Southern Illinois University Hilbert Spaces: InfiniteDimensional Vector Spaces October
More information10701/ Recitation : Linear Algebra Review (based on notes written by Jing Xiang)
10701/15781 Recitation : Linear Algebra Review (based on notes written by Jing Xiang) Manojit Nandi February 1, 2014 Outline Linear Algebra General Properties Matrix Operations Inner Products and Orthogonal
More informationECS130 Scientific Computing. Lecture 1: Introduction. Monday, January 7, 10:00 10:50 am
ECS130 Scientific Computing Lecture 1: Introduction Monday, January 7, 10:00 10:50 am About Course: ECS130 Scientific Computing Professor: Zhaojun Bai Webpage: http://web.cs.ucdavis.edu/~bai/ecs130/ Today
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationT.8. PerronFrobenius theory of positive matrices From: H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003
T.8. PerronFrobenius theory of positive matrices From: H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003 A vector x R n is called positive, symbolically x > 0,
More informationMATH 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 informationLINEAR 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 ntuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication
More informationLINEAR 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 informationSpectral theory for compact operators on Banach spaces
68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy
More informationLecture notes on Quantum Computing. Chapter 1 Mathematical Background
Lecture notes on Quantum Computing Chapter 1 Mathematical Background Vector states of a quantum system with n physical states are represented by unique vectors in C n, the set of n 1 column vectors 1 For
More information1. 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 informationFunctional Analysis Review
Functional Analysis Review Lorenzo Rosasco slides courtesy of Andre Wibisono 9.520: Statistical Learning Theory and Applications September 9, 2013 1 2 3 4 Vector Space A vector space is a set V with binary
More informationMath 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 informationMath 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 78pm in 117 MacMillan What will be covered The exam will cover material from the lectures
More informationProblem # Max points possible Actual score Total 120
FINAL EXAMINATION  MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to
More informationReview of similarity transformation and Singular Value Decomposition
Review of similarity transformation and Singular Value Decomposition Nasser M Abbasi Applied Mathematics Department, California State University, Fullerton July 8 7 page compiled on June 9, 5 at 9:5pm
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data TwoDimensional Plots Programming in
More information22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices
m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix
More information