Large Scale Data Analysis Using Deep Learning

Size: px
Start display at page:

Download "Large Scale Data Analysis Using Deep Learning"

Transcription

1 Large Scale Data Analysis Using Deep Learning Linear Algebra U Kang Seoul National University U Kang 1

2 In This Lecture Overview of linear algebra (but, not a comprehensive survey) Focused on the subset most relevant to deep learning For details on linear algebra, refer to Linear Algebra and Its Applications by Gilbert Strang U Kang 2

3 Scalar A scalar is a single number Integers, real number, rational numbers, etc. We denote it with italic font a, n, x U Kang 3

4 Vectors A vector is a 1-D array of numbers Can be real, binary, integer, etc. Notation for type and size U Kang 4

5 Matrices A matrix is a 2-D array of numbers Example notation for type and shape U Kang 5

6 Tensors A tensor is an array of numbers, that may have Zero dimensions, and be a scalar One dimension, and be a vector Two dimensions, and be a matrix Three dimensions or more Matrix over time Knowledge base (subject, verb, object) U Kang 6

7 Matrix Transpose (A T ) i,j = A j,i (AB) T = B T A T U Kang 7

8 Matrix Product C = AB U Kang 8

9 Matrix Product Matrix product as sum of outer product Product with a diagonal matrix U Kang 9

10 Identity Matrix Example identity matrix: I 3 U Kang 10

11 Systems of Equations expands to U Kang 11

12 Solving Systems of Equations A linear system of equations can have: No solution 3x + 2y = 6, 3x + 2y = 12 Many solutions 3x + 2y = 6, 6x + 4y = 12 Exactly one solution: this means multiplication by the matrix is an invertible function AAAA = bb xx = AA 1 bb U Kang 12

13 Matrix Inversion Matrix inverse: an inverse A -1 of an nxn matrix A satisfies AAAA 1 = AA 1 AA = II nn Solving a system using an inverse U Kang 13

14 Invertibility Matrix A cannot be inverted if More rows than columns More columns than rows Redundant rows/columns ( linearly dependent, low rank ) full rank The number 0 is an eigenvalue of A A non-invertible matrix is called a singular matrix An invertible matrix is called non-singular U Kang 14

15 Linear Dependence and Span Linear combination of vectors {vv 1,, vv nn } : ii cc ii vv (ii) The span of a set of vectors is the set of all points obtainable by linear combination of the original vectors Matrix-vector product Ax can be viewed as a linear combination of column vectors of A: AAAA = ii xx ii AA :,ii The span of columns of A is called column space or range of A A set of vectors is linearly independent if no vector in the set is a linear combination of the other vectors linearly dependent U Kang 15

16 Rank of a Matrix Rank of a matrix A: Number of linearly independent columns (or rows) of A For example: Matrix A = s rank? Why? U Kang 16

17 Norms Functions that measure how large a vector is Similar to a distance between zero and the point represented by the vector Formally, a norm is any function f that satisfies the following: U Kang 17

18 Norms L p norm Most popular norm: L2 norm, p = 2 Euclidean distance For matrix/tensor, Frobenius norm (L F ) does the same thing L1 norm, p = 1: Called Manhattan distance Max norm, infinite p: Dot product in terms of norm: U Kang 18

19 Special Matrices and Vectors Diagonal matrix 2 by 2 : aa bb We use diag(v) to denote the diagonal matrix where v is the vector containing the diagonal elements Inverse of diagonal matrix is computed easily U Kang 19

20 Special Matrices and Vectors Unit vector Symmetric matrix Orthogonal matrix U Kang 20

21 Eigenvector and Eigenvalue Eigenvector and eigenvalue of A Eigenvalue/eigenvector pairs are defined only for square matrix AA RR nn nn # of eigenvalues: n There may be duplicates Eigenvalues and eigenvectors can contain real or imaginary numbers U Kang 21

22 Intuition A as vector transformation x 2 1 A x = x x U Kang 22

23 Intuition By defn., eigenvectors remain parallel to them selves ( fixed points ) λ * v = A v U Kang 23

24 Eigendecomposition Eigendecomposition of a matrix Let A be a square (n x n) matrix with n linearly independent eigenvectors (=diagonizable) Then A can be factorized as where V is an (nxn) matrix whose i th column is the i th eigenvector of A U Kang 24

25 Eigendecomposition Every real symmetric matrix has a real, orthogonal eigendecomposition A real orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (orthonormal vectors) QQ TT QQ = QQ QQ TT = II QQ TT = QQ 1 U Kang 25

26 Eigendecomposition Interpreting matrix-vector multiplication Ax using eigendecomposition U Kang 26

27 Eigendecomposition Understanding optimization of ff xx = xx TT AAAA, s.t. xx 2 = 1, using eigenvalues and eigenvectors U Kang 27

28 Eigendecomposition Positive definite matrix: a real symmetric matrix whose eigenvalues are all positive Positive semidefinite matrix Negative definite matrix Negative semidefinite matrix For positive semidefinite matrix A, xx, xx TT AAAA 0. For positive definite matrix A, xx 0, xx TT AAAA > 0 U Kang 28

29 Singular Value Decomposition (SVD) Similar to eigendecomposition More general; matrix need not be square AA = UUΛVV TT U Kang 29

30 SVD - Definition A [n x m] = U [n x r] Λ [ r x r] (V [m x r] ) T A: n x m matrix (eg., n documents, m terms) U: n x r matrix (n documents, r concepts) Left singular vectors Λ: r x r diagonal matrix (strength of each concept ) (r : rank of the matrix) V: m x r matrix (m terms, r concepts) Right singular vectors U Kang 30

31 SVD - Definition A [n x m] = U [n x r] Λ [ r x r] (V [m x r] ) T A U Λ = x x r x r V T r x m n x m n x r U Kang 31

32 SVD - Properties THEOREM [Press+92]: always possible to decompose matrix A into A = U Λ V T, where U, Λ, V: unique (*) U, V: column orthonormal (ie., columns are unit vectors, orthogonal to each other) U T U = I; V T V = I (I: identity matrix) Λ: singular are positive, and sorted in decreasing order U Kang 32

33 SVD - Properties AA UUΛVV TT = σσ ii (uu ii vv TT ii ) ii n n m A m ΛΛ V T U Best rank-k approximation in L F U Kang 33

34 SVD and eigendecomposition The left-singular vectors of A are the eigenvectors of AA T The right-singular vectors of A are the eigenvectors of A T A The non-zero singular values of A are the square root of the eigenvalues of A T A U Kang 34

35 Moore-Penrose Pseudoinverse Assume we want to solve A x = y for x What if A is not invertible? What if A is not square? Still, we can find the best x by using pseudoinverse For an (n x m) matrix A, the pseudoinverse A + is an (m x n) matrix U Kang 35

36 Moore-Penrose Pseudoinverse If the equation has: Exactly one solution: this is the same as the inverse No solution: this gives us the solution with the smallest error Ax y 2 Over-specified case Many solutions: this gives us the solution with the smallest norm of x Under-specified case U Kang 36

37 Pseudoinverse: Over-specified case No solution: this gives us the solution with the smallest error Ax y 2 [3 2] T [x] = [1 2] T (i.e., 3x = 1, 2x = 2) 2 1 desirable point y reachable points (3x, 2x) ([3 2] T ) + [1 2] = = 7/13 This method is called the least square U Kang 37

38 Pseudoinverse: Under-specified case Many solutions: this gives us the solution with the smallest norm of x [1 2] [w z] T = 4 (i.e., 1 w + 2 z = 4) 2 z 1 x 0 shortest-length solution all possible solutions w [1 2] + 4 = [1/5 2/5] T 4 = [4/5 8/5] U Kang 38

39 Computing the Pseudoinverse The SVD allows the computation of the pseudoinverse: U Kang 39

40 Trace TTTT AA = ii AA ii,ii TTTT AAAAAA = TTTT CCCCCC = TTTT(BBBBBB) TTTT AA = TTTT(AA TT ) AA FF = TTTT(AAAA TT ) For scalar a, TTTT aa = aa The trace of a matrix is also computed by the sum of all eigenvalues U Kang 40

41 What you need to know Linear algebra is crucial for understanding notations and mechanisms of many ML/deep learning methods Important concepts Matrix product, identity, invertibility, norm, eigendecomposition, singular value decomposition, pseudo inverse, and trace U Kang 41

42 Questions? U Kang 42

Linear Algebra for Machine Learning. Sargur N. Srihari

Linear Algebra for Machine Learning. Sargur N. Srihari Linear Algebra for Machine Learning Sargur N. srihari@cedar.buffalo.edu 1 Overview Linear Algebra is based on continuous math rather than discrete math Computer scientists have little experience with it

More information

Deep Learning Book Notes Chapter 2: Linear Algebra

Deep Learning Book Notes Chapter 2: Linear Algebra Deep Learning Book Notes Chapter 2: Linear Algebra Compiled By: Abhinaba Bala, Dakshit Agrawal, Mohit Jain Section 2.1: Scalars, Vectors, Matrices and Tensors Scalar Single Number Lowercase names in italic

More information

Linear Algebra (Review) Volker Tresp 2018

Linear Algebra (Review) Volker Tresp 2018 Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A one-dimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the i-th component of c c T = (c 1, c

More information

Review of Linear Algebra

Review 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 information

Linear Algebra (Review) Volker Tresp 2017

Linear Algebra (Review) Volker Tresp 2017 Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.

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

Linear Algebra - Part II

Linear 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 information

Image Registration Lecture 2: Vectors and Matrices

Image Registration Lecture 2: Vectors and Matrices Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this

More information

Linear Algebra. Shan-Hung Wu. Department of Computer Science, National Tsing Hua University, Taiwan. Large-Scale ML, Fall 2016

Linear Algebra. Shan-Hung Wu. Department of Computer Science, National Tsing Hua University, Taiwan. Large-Scale ML, Fall 2016 Linear Algebra Shan-Hung Wu shwu@cs.nthu.edu.tw Department of Computer Science, National Tsing Hua University, Taiwan Large-Scale ML, Fall 2016 Shan-Hung Wu (CS, NTHU) Linear Algebra Large-Scale ML, Fall

More information

Lecture II: Linear Algebra Revisited

Lecture 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 information

UNIT 6: The singular value decomposition.

UNIT 6: The singular value decomposition. UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T

More information

Maths for Signals and Systems Linear Algebra in Engineering

Maths for Signals and Systems Linear Algebra in Engineering Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE

More information

Matrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A =

Matrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = 30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional 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 information

Foundations of Computer Vision

Foundations of Computer Vision Foundations of Computer Vision Wesley. E. Snyder North Carolina State University Hairong Qi University of Tennessee, Knoxville Last Edited February 8, 2017 1 3.2. A BRIEF REVIEW OF LINEAR ALGEBRA Apply

More information

Properties of Matrices and Operations on Matrices

Properties 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 information

1 Linearity and Linear Systems

1 Linearity and Linear Systems Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 7-8 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)

More information

The study of linear algebra involves several types of mathematical objects:

The study of linear algebra involves several types of mathematical objects: Chapter 2 Linear Algebra Linear algebra is a branch of mathematics that is widely used throughout science and engineering. However, because linear algebra is a form of continuous rather than discrete mathematics,

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

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

CS 246 Review of Linear Algebra 01/17/19

CS 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 information

Dimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas

Dimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas Dimensionality Reduction: PCA Nicholas Ruozzi University of Texas at Dallas Eigenvalues λ is an eigenvalue of a matrix A R n n if the linear system Ax = λx has at least one non-zero solution If Ax = λx

More information

Introduction to Data Mining

Introduction to Data Mining Introduction to Data Mining Lecture #21: Dimensionality Reduction Seoul National University 1 In This Lecture Understand the motivation and applications of dimensionality reduction Learn the definition

More information

Mathematical foundations - linear algebra

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 information

Singular Value Decomposition

Singular Value Decomposition Singular Value Decomposition Motivatation The diagonalization theorem play a part in many interesting applications. Unfortunately not all matrices can be factored as A = PDP However a factorization A =

More information

Econ Slides from Lecture 7

Econ 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 information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Dr Gerhard Roth COMP 40A Winter 05 Version Linear algebra Is an important area of mathematics It is the basis of computer vision Is very widely taught, and there are many resources

More information

Linear Algebra Formulas. Ben Lee

Linear Algebra Formulas. Ben Lee Linear Algebra Formulas Ben Lee January 27, 2016 Definitions and Terms Diagonal: Diagonal of matrix A is a collection of entries A ij where i = j. Diagonal Matrix: A matrix (usually square), where entries

More information

CS 143 Linear Algebra Review

CS 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 information

Review of Some Concepts from Linear Algebra: Part 2

Review 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 information

Chapter 3 Transformations

Chapter 3 Transformations Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta 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 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

[POLS 8500] Review of Linear Algebra, Probability and Information Theory

[POLS 8500] Review of Linear Algebra, Probability and Information Theory [POLS 8500] Review of Linear Algebra, Probability and Information Theory Professor Jason Anastasopoulos ljanastas@uga.edu January 12, 2017 For today... Basic linear algebra. Basic probability. Programming

More information

7. Symmetric Matrices and Quadratic Forms

7. Symmetric Matrices and Quadratic Forms Linear Algebra 7. Symmetric Matrices and Quadratic Forms CSIE NCU 1 7. Symmetric Matrices and Quadratic Forms 7.1 Diagonalization of symmetric matrices 2 7.2 Quadratic forms.. 9 7.4 The singular value

More information

2. Review of Linear Algebra

2. 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 information

j=1 u 1jv 1j. 1/ 2 Lemma 1. An orthogonal set of vectors must be linearly independent.

j=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 information

Background Mathematics (2/2) 1. David Barber

Background Mathematics (2/2) 1. David Barber Background Mathematics (2/2) 1 David Barber University College London Modified by Samson Cheung (sccheung@ieee.org) 1 These slides accompany the book Bayesian Reasoning and Machine Learning. The book and

More information

Singular Value Decomposition. 1 Singular Value Decomposition and the Four Fundamental Subspaces

Singular Value Decomposition. 1 Singular Value Decomposition and the Four Fundamental Subspaces Singular Value Decomposition This handout is a review of some basic concepts in linear algebra For a detailed introduction, consult a linear algebra text Linear lgebra and its pplications by Gilbert Strang

More information

Assignment 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. 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 information

Linear Algebra: Characteristic Value Problem

Linear Algebra: Characteristic Value Problem Linear Algebra: Characteristic Value Problem . The Characteristic Value Problem Let < be the set of real numbers and { be the set of complex numbers. Given an n n real matrix A; does there exist a number

More information

linearly indepedent eigenvectors as the multiplicity of the root, but in general there may be no more than one. For further discussion, assume matrice

linearly indepedent eigenvectors as the multiplicity of the root, but in general there may be no more than one. For further discussion, assume matrice 3. Eigenvalues and Eigenvectors, Spectral Representation 3.. Eigenvalues and Eigenvectors A vector ' is eigenvector of a matrix K, if K' is parallel to ' and ' 6, i.e., K' k' k is the eigenvalue. If is

More information

Lecture 3: Review of Linear Algebra

Lecture 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 information

Lecture 3: Review of Linear Algebra

Lecture 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 information

Functional Analysis Review

Functional 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 information

14 Singular Value Decomposition

14 Singular Value Decomposition 14 Singular Value Decomposition For any high-dimensional data analysis, one s first thought should often be: can I use an SVD? The singular value decomposition is an invaluable analysis tool for dealing

More information

Appendix A: Matrices

Appendix A: Matrices Appendix A: Matrices A matrix is a rectangular array of numbers Such arrays have rows and columns The numbers of rows and columns are referred to as the dimensions of a matrix A matrix with, say, 5 rows

More information

Mathematical foundations - linear algebra

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 information

Basic Calculus Review

Basic 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 information

Data Preprocessing. Jilles Vreeken IRDM 15/ Oct 2015

Data Preprocessing. Jilles Vreeken IRDM 15/ Oct 2015 Data Preprocessing Jilles Vreeken 22 Oct 2015 So, how do you pronounce Jilles Yill-less Vreeken Fray-can Okay, now we can talk. Questions of the day How do we preprocess data before we can extract anything

More information

Matrix Algebra: Summary

Matrix 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 information

B553 Lecture 5: Matrix Algebra Review

B553 Lecture 5: Matrix Algebra Review B553 Lecture 5: Matrix Algebra Review Kris Hauser January 19, 2012 We have seen in prior lectures how vectors represent points in R n and gradients of functions. Matrices represent linear transformations

More information

Linear Algebra. Lecture slides for Chapter 2 of Deep Learning Ian Goodfellow

Linear Algebra. Lecture slides for Chapter 2 of Deep Learning Ian Goodfellow Linear lgebra Lecture slides for Chapter 2 of Deep Learning Ian Goodfellow 206-06-24 bout this chapter Not a comprehensive survey of all of linear algebra Focused on the subset most relevant to deep learning

More information

Linear Algebra and Eigenproblems

Linear Algebra and Eigenproblems Appendix A A Linear Algebra and Eigenproblems A working knowledge of linear algebra is key to understanding many of the issues raised in this work. In particular, many of the discussions of the details

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

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

Stat 159/259: Linear Algebra Notes

Stat 159/259: Linear Algebra Notes Stat 159/259: Linear Algebra Notes Jarrod Millman November 16, 2015 Abstract These notes assume you ve taken a semester of undergraduate linear algebra. In particular, I assume you are familiar with the

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

3 (Maths) Linear Algebra

3 (Maths) Linear Algebra 3 (Maths) Linear Algebra References: Simon and Blume, chapters 6 to 11, 16 and 23; Pemberton and Rau, chapters 11 to 13 and 25; Sundaram, sections 1.3 and 1.5. The methods and concepts of linear algebra

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

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

Basic Concepts in Linear Algebra

Basic Concepts in Linear Algebra Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear

More information

Fall TMA4145 Linear Methods. Exercise set Given the matrix 1 2

Fall 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 information

Singular Value Decomposition

Singular Value Decomposition Chapter 6 Singular Value Decomposition In Chapter 5, we derived a number of algorithms for computing the eigenvalues and eigenvectors of matrices A R n n. Having developed this machinery, we complete our

More information

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

More information

Eigenvalues and diagonalization

Eigenvalues and diagonalization Eigenvalues and diagonalization Patrick Breheny November 15 Patrick Breheny BST 764: Applied Statistical Modeling 1/20 Introduction The next topic in our course, principal components analysis, revolves

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

Singular Value Decompsition

Singular Value Decompsition Singular Value Decompsition Massoud Malek One of the most useful results from linear algebra, is a matrix decomposition known as the singular value decomposition It has many useful applications in almost

More information

Lecture notes on Quantum Computing. Chapter 1 Mathematical Background

Lecture 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 information

Review of Basic Concepts in Linear Algebra

Review of Basic Concepts in Linear Algebra Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra

More information

Introduction to Numerical Linear Algebra II

Introduction to Numerical Linear Algebra II Introduction to Numerical Linear Algebra II Petros Drineas These slides were prepared by Ilse Ipsen for the 2015 Gene Golub SIAM Summer School on RandNLA 1 / 49 Overview We will cover this material in

More information

10-701/ Recitation : Linear Algebra Review (based on notes written by Jing Xiang)

10-701/ Recitation : Linear Algebra Review (based on notes written by Jing Xiang) 10-701/15-781 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 information

Singular Value Decomposition (SVD)

Singular Value Decomposition (SVD) School of Computing National University of Singapore CS CS524 Theoretical Foundations of Multimedia More Linear Algebra Singular Value Decomposition (SVD) The highpoint of linear algebra Gilbert Strang

More information

Lecture 2: Linear Algebra

Lecture 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 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

LinGloss. A glossary of linear algebra

LinGloss. A glossary of linear algebra LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal

More information

Linear Algebra Review

Linear 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 information

Lecture 6 Positive Definite Matrices

Lecture 6 Positive Definite Matrices Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices

More information

The Singular Value Decomposition

The Singular Value Decomposition The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will

More information

COMP 558 lecture 18 Nov. 15, 2010

COMP 558 lecture 18 Nov. 15, 2010 Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to

More information

1. 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

1. 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 information

b 1 b 2.. b = b m A = [a 1,a 2,...,a n ] where a 1,j a 2,j a j = a m,j Let A R m n and x 1 x 2 x = x n

b 1 b 2.. b = b m A = [a 1,a 2,...,a n ] where a 1,j a 2,j a j = a m,j Let A R m n and x 1 x 2 x = x n Lectures -2: Linear Algebra Background Almost all linear and nonlinear problems in scientific computation require the use of linear algebra These lectures review basic concepts in a way that has proven

More information

EE731 Lecture Notes: Matrix Computations for Signal Processing

EE731 Lecture Notes: Matrix Computations for Signal Processing EE731 Lecture Notes: Matrix Computations for Signal Processing James P. Reilly c Department of Electrical and Computer Engineering McMaster University September 22, 2005 0 Preface This collection of ten

More information

Data Mining Lecture 4: Covariance, EVD, PCA & SVD

Data Mining Lecture 4: Covariance, EVD, PCA & SVD Data Mining Lecture 4: Covariance, EVD, PCA & SVD Jo Houghton ECS Southampton February 25, 2019 1 / 28 Variance and Covariance - Expectation A random variable takes on different values due to chance The

More information

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2017 LECTURE 5

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2017 LECTURE 5 STAT 39: MATHEMATICAL COMPUTATIONS I FALL 17 LECTURE 5 1 existence of svd Theorem 1 (Existence of SVD) Every matrix has a singular value decomposition (condensed version) Proof Let A C m n and for simplicity

More information

Prof. Dr.-Ing. Armin Dekorsy Department of Communications Engineering. Stochastic Processes and Linear Algebra Recap Slides

Prof. Dr.-Ing. Armin Dekorsy Department of Communications Engineering. Stochastic Processes and Linear Algebra Recap Slides Prof. Dr.-Ing. Armin Dekorsy Department of Communications Engineering Stochastic Processes and Linear Algebra Recap Slides Stochastic processes and variables XX tt 0 = XX xx nn (tt) xx 2 (tt) XX tt XX

More information

EIGENVALUES AND SINGULAR VALUE DECOMPOSITION

EIGENVALUES AND SINGULAR VALUE DECOMPOSITION APPENDIX B EIGENVALUES AND SINGULAR VALUE DECOMPOSITION B.1 LINEAR EQUATIONS AND INVERSES Problems of linear estimation can be written in terms of a linear matrix equation whose solution provides the required

More information

DS-GA 1002 Lecture notes 10 November 23, Linear models

DS-GA 1002 Lecture notes 10 November 23, Linear models DS-GA 2 Lecture notes November 23, 2 Linear functions Linear models A linear model encodes the assumption that two quantities are linearly related. Mathematically, this is characterized using linear functions.

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

Designing Information Devices and Systems II

Designing Information Devices and Systems II EECS 16B Fall 2016 Designing Information Devices and Systems II Linear Algebra Notes Introduction In this set of notes, we will derive the linear least squares equation, study the properties symmetric

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

Problem # Max points possible Actual score Total 120

Problem # 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 information

Lecture 02 Linear Algebra Basics

Lecture 02 Linear Algebra Basics Introduction to Computational Data Analysis CX4240, 2019 Spring Lecture 02 Linear Algebra Basics Chao Zhang College of Computing Georgia Tech These slides are based on slides from Le Song and Andres Mendez-Vazquez.

More information

Quick Tour of Linear Algebra and Graph Theory

Quick Tour of Linear Algebra and Graph Theory Quick Tour of Linear Algebra and Graph Theory CS224W: Social and Information Network Analysis Fall 2014 David Hallac Based on Peter Lofgren, Yu Wayne Wu, and Borja Pelato s previous versions Matrices and

More information

2. LINEAR ALGEBRA. 1. Definitions. 2. Linear least squares problem. 3. QR factorization. 4. Singular value decomposition (SVD) 5.

2. LINEAR ALGEBRA. 1. Definitions. 2. Linear least squares problem. 3. QR factorization. 4. Singular value decomposition (SVD) 5. 2. LINEAR ALGEBRA Outline 1. Definitions 2. Linear least squares problem 3. QR factorization 4. Singular value decomposition (SVD) 5. Pseudo-inverse 6. Eigenvalue decomposition (EVD) 1 Definitions Vector

More information

Cheat Sheet for MATH461

Cheat Sheet for MATH461 Cheat Sheet for MATH46 Here is the stuff you really need to remember for the exams Linear systems Ax = b Problem: We consider a linear system of m equations for n unknowns x,,x n : For a given matrix A

More information

Problem Set 1. Homeworks will graded based on content and clarity. Please show your work clearly for full credit.

Problem 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 information

Jeffrey D. Ullman Stanford University

Jeffrey D. Ullman Stanford University Jeffrey D. Ullman Stanford University 2 Often, our data can be represented by an m-by-n matrix. And this matrix can be closely approximated by the product of two matrices that share a small common dimension

More information

2. Every linear system with the same number of equations as unknowns has a unique solution.

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

More information

Chapter 1. Matrix Algebra

Chapter 1. Matrix Algebra ST4233, Linear Models, Semester 1 2008-2009 Chapter 1. Matrix Algebra 1 Matrix and vector notation Definition 1.1 A matrix is a rectangular or square array of numbers of variables. We use uppercase boldface

More information

Chap 3. Linear Algebra

Chap 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 information

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)

MATH 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 information