Linear Algebra & Geometry why is linear algebra useful in computer vision?


 Ross Hall
 1 years ago
 Views:
Transcription
1 Linear Algebra & Geometry why is linear algebra useful in computer vision? References: Any book on linear algebra! [HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia I. Camps, Penn State University
2 Why is linear algebra useful in computer vision? Representation 3D points in the scene 2D points in the image Coordinates will be used to Perform geometrical transformations Associate 3D with 2D points Images are matrices of numbers Find properties of these numbers
3 Agenda 1. Basics definitions and properties 2. Geometrical transformations 3. SVD and its applications
4 Vectors (i.e., 2D or 3D vectors) P = [x,y,z] p = [x,y] 3D world Image
5 Vectors (i.e., 2D vectors) x2 v θ P Magnitude: x1 If, Is a UNIT vector Is a unit vector Orientation:
6 Vector Addition v+w v w
7 Vector Subtraction v vw w
8 Scalar Product v av
9 Inner (dot) Product v α w The inner product is a SCALAR!
10 Orthonormal Basis x2 j i v θ P x1
11 Vector (cross) Product u w α v The cross product is a VECTOR! Magnitude:kuk = kv wk = kvkkwk sin Orientation:
12 Vector Product Computation i = (1,0,0) j = (0,1,0) k = (0,0,1) i = 1 j = 1 k = 1 i = j k j = k i k = i j = ( x2 y 3 x3 y ) i + 2 ( x3 y 1 x1 y3) j+ ( x1 y 2 x2 y1 ) k
13 Matrices A n m = 2 3 a 11 a a 1m a 21 a a 2m a n1 a n2... a nm Pixel s intensity value Sum: A and B must have the same dimensions! Example:
14 Matrices A n m = 2 3 a 11 a a 1m a 21 a a 2m a n1 a n2... a nm Product: A and B must have compatible dimensions!
15 Matrix Transpose Definition: C m n = A T n m c ij = a ji Identities: (A + B) T = A T + B T (AB) T = B T A T If A = A T, then A is symmetric
16 Matrix Determinant Useful value computed from the elements of a square matrix A det [ a 11 ] = a11 [ ] a11 a det 12 = a a 21 a 11 a 22 a 12 a a 11 a 12 a 13 det a 21 a 22 a 23 = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 31 a 32 a 33 a 13 a 22 a 31 a 23 a 32 a 11 a 33 a 12 a 21
17 Matrix Inverse Does not exist for all matrices, necessary (but not sufficient) that the matrix is square AA 1 = A 1 A = I [ ] 1 A 1 a11 a = 12 = 1 a 21 a 22 det A If det A = 0, A does not have an inverse. [ a22 a 12 a 21 a 11 ], det A 0
18 2D Geometrical Transformations
19 2D Translation P P t
20 2D Translation Equation ty y P t P x tx
21 2D Translation using Matrices ty P t P y x tx
22 Homogeneous Coordinates Multiply the coordinates by a nonzero scalar and add an extra coordinate equal to that scalar. For example,
23 Back to Cartesian Coordinates: Divide by the last coordinate and eliminate it. For example, NOTE: in our example the scalar was 1
24 2D Translation using Homogeneous Coordinates ty P t P y t P x tx
25 Scaling P P
26 Scaling Equation s y y P y P x s x x
27 Scaling & Translating P P P =S P P =T P P =T P =T (S P)=(T S) P = A P
28 Scaling & Translating A
29 Translating & Scaling = Scaling & Translating?
30 Rotation P P
31 Rotation Equations Counterclockwise rotation by an angle θ y y P x θ x P
32 Degrees of Freedom R is 2x2 4 elements Note: R belongs to the category of normal matrices and satisfies many interesting properties:
33 Rotation+ Scaling +Translation P = (T R S) P If s x =s y, this is a similarity transformation!
34 Transformation in 2D Isometries Similarities Affinity Projective
35 Transformation in 2D Isometries: [Euclideans]  Preserve distance (areas)  3 DOF  Regulate motion of rigid object
36 Transformation in 2D Similarities:  Preserve  ratio of lengths  angles 4 DOF
37 Affinities: Transformation in 2D
38 Transformation in 2D Affinities: Preserve:  Parallel lines  Ratio of areas  Ratio of lengths on collinear lines  others  6 DOF
39 Transformation in 2D Projective:  8 DOF  Preserve:  cross ratio of 4 collinear points  collinearity  and a few others
40 Eigenvalues and Eigenvectors A eigenvalue λ and eigenvector u satisfies where A is a square matrix. Au = λu Multiplying u by A scales u by λ
41 Eigenvalues and Eigenvectors Rearranging the previous equation gives the system Au λu = (A λi)u = 0 which has a solution if and only if det(a λi) = 0. The eigenvalues are the roots of this determinant which is polynomial in λ. Substitute the resulting eigenvalues back into Au = λu and solve to obtain the corresponding eigenvector.
42 Singular Value Decomposition UΣV T = A Where U and V are orthogonal matrices, and Σ is a diagonal matrix. For example:
43 Singular Value decomposition that
44 An Numerical Example Look at how the multiplication works out, left to right: Column 1 of U gets scaled by the first value from Σ. The resulting vector gets scaled by row 1 of V T to produce a contribution to the columns of A
45 An Numerical Example + = Each product of (column i of U) (value i from Σ) (row i of V T ) produces a
46 An Numerical Example We can look at Σ to see that the first column has a large effect while the second column has a much smaller effect in this example
47 SVD Applications For this image, using only the first 10 of 300 singular values produces a recognizable reconstruction So, SVD can be used for image compression
48 Principal Component Analysis Remember, columns of U are the Principal Components of the data: the major patterns that can be added to produce the columns of the original matrix One use of this is to construct a matrix where each column is a separate data sample Run SVD on that matrix, and look at the first few columns of U to see patterns that are common among the columns This is called Principal Component Analysis (or PCA) of the data samples
49 Principal Component Analysis Often, raw data samples have a lot of redundancy and patterns PCA can allow you to represent data samples as weights on the principal components, rather than using the original raw form of the data By representing each sample as just those weights, you can represent just the meat of what s different between samples. This minimal representation makes machine learning and other algorithms much more efficient
50
51
Linear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: Any book on linear algebra! [HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationLecture: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab Lecture 111 Recap  Curse of dimensionality Assume 5000 points uniformly distributed
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationImage 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 informationSingular 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 informationReview 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 information1 Last time: leastsquares problems
MATH Linear algebra (Fall 07) Lecture Last time: leastsquares problems Definition. If A is an m n matrix and b R m, then a leastsquares solution to the linear system Ax = b is a vector x R n such that
More informationand let s calculate the image of some vectors under the transformation T.
Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =
More informationLinear 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 informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationThe Singular Value Decomposition (SVD) and Principal Component Analysis (PCA)
Chapter 5 The Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) 5.1 Basics of SVD 5.1.1 Review of Key Concepts We review some key definitions and results about matrices that will
More informationBackground 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 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 informationMaths 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 informationMath 215 HW #9 Solutions
Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith
More informationLinear Algebra V = T = ( 4 3 ).
Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a dimensional column vector and V is a 5dimensional
More informationLinear 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 ToolBox Linear Equation Systems Discretization of differential equations: solving linear equations
More informationPositive Definite Matrix
1/29 ChiaPing Chen Professor Department of Computer Science and Engineering National Sun Yatsen University Linear Algebra Positive Definite, Negative Definite, Indefinite 2/29 Pure Quadratic Function
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 10: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11)
Lecture 1: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11) The eigenvalue problem, Ax= λ x, occurs in many, many contexts: classical mechanics, quantum mechanics, optics 22 Eigenvectors and
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 informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationLecture 3: Matrix and Matrix Operations
Lecture 3: Matrix and Matrix Operations Representation, row vector, column vector, element of a matrix. Examples of matrix representations Tables and spreadsheets ScalarMatrix operation: Scaling a matrix
More information(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =
. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)
More informationExtra Problems for Math 2050 Linear Algebra I
Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as
More informationDATA MINING LECTURE 8. Dimensionality Reduction PCA  SVD
DATA MINING LECTURE 8 Dimensionality Reduction PCA  SVD The curse of dimensionality Real data usually have thousands, or millions of dimensions E.g., web documents, where the dimensionality is the vocabulary
More informationComputational math: Assignment 1
Computational math: Assignment 1 Thanks Ting Gao for her Latex file 11 Let B be a 4 4 matrix to which we apply the following operations: 1double column 1, halve row 3, 3add row 3 to row 1, 4interchange
More informationFundamentals of Matrices
Maschinelles Lernen II Fundamentals of Matrices Christoph Sawade/Niels Landwehr/Blaine Nelson Tobias Scheffer Matrix Examples Recap: Data Linear Model: f i x = w i T x Let X = x x n be the data matrix
More informationLinear 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 informationQuiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown.
Quiz 1) Simplify 9999 999 9999 998 9999 998 2) Locate your 1 st order neighbors Name Hometown Me Name Hometown Name Hometown Name Hometown Solving Linear Algebraic Equa3ons Basic Concepts Here only real
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 informationLecture 6 Positive Definite Matrices
Linear Algebra Lecture 6 Positive Definite Matrices Prof. ChunHung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices
More informationVectors and Matrices Statistics with Vectors and Matrices
Vectors and Matrices Statistics with Vectors and Matrices Lecture 3 September 7, 005 Analysis Lecture #39/7/005 Slide 1 of 55 Today s Lecture Vectors and Matrices (Supplement A  augmented with SAS proc
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 informationCOMP 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 informationLinear Algebra and Matrices
Linear Algebra and Matrices 4 Overview In this chapter we studying true matrix operations, not element operations as was done in earlier chapters. Working with MAT LAB functions should now be fairly routine.
More informationComputational Methods. Eigenvalues and Singular Values
Computational Methods Eigenvalues and Singular Values Manfred Huber 2010 1 Eigenvalues and Singular Values Eigenvalues and singular values describe important aspects of transformations and of data relations
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationMatrices and Deformation
ES 111 Mathematical Methods in the Earth Sciences Matrices and Deformation Lecture Outline 13  Thurs 9th Nov 2017 Strain Ellipse and Eigenvectors One way of thinking about a matrix is that it operates
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 informationMatrices and Matrix Algebra.
Matrices and Matrix Algebra 3.1. Operations on Matrices Matrix Notation and Terminology Matrix: a rectangular array of numbers, called entries. A matrix with m rows and n columns m n A n n matrix : a square
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 informationAM 205: lecture 8. Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition
AM 205: lecture 8 Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition QR Factorization A matrix A R m n, m n, can be factorized
More informationEXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants
EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For
More informationANSWERS. E k E 2 E 1 A = B
MATH 7 Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,
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 informationStat 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 informationApplied Mathematics 205. Unit II: Numerical Linear Algebra. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit II: Numerical Linear Algebra Lecturer: Dr. David Knezevic Unit II: Numerical Linear Algebra Chapter II.3: QR Factorization, SVD 2 / 66 QR Factorization 3 / 66 QR Factorization
More informationFace Recognition and Biometric Systems
The Eigenfaces method Plan of the lecture Principal Components Analysis main idea Feature extraction by PCA face recognition Eigenfaces training feature extraction Literature M.A.Turk, A.P.Pentland Face
More informationVector Spaces, Affine Spaces, and Metric Spaces
Vector Spaces, Affine Spaces, and Metric Spaces 2 This chapter is only meant to give a short overview of the most important concepts in linear algebra, affine spaces, and metric spaces and is not intended
More information18.06SC Final Exam Solutions
18.06SC Final Exam Solutions 1 (4+7=11 pts.) Suppose A is 3 by 4, and Ax = 0 has exactly 2 special solutions: 1 2 x 1 = 1 and x 2 = 1 1 0 0 1 (a) Remembering that A is 3 by 4, find its row reduced echelon
More informationPrincipal Component Analysis
Principal Component Analysis CS5240 Theoretical Foundations in Multimedia Leow Wee Kheng Department of Computer Science School of Computing National University of Singapore Leow Wee Kheng (NUS) Principal
More informationReview of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem
Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem Steven J. Miller June 19, 2004 Abstract Matrices can be thought of as rectangular (often square) arrays of numbers, or as
More informationMATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra
MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;
More information4. Determinants.
4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.
More informationLecture 2: Linear Algebra Review
EE 227A: Convex Optimization and Applications January 19 Lecture 2: Linear Algebra Review Lecturer: Mert Pilanci Reading assignment: Appendix C of BV. Sections 26 of the web textbook 1 2.1 Vectors 2.1.1
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 informationMathematical Foundations
Chapter 1 Mathematical Foundations 1.1 BigO Notations In the description of algorithmic complexity, we often have to use the order notations, often in terms of big O and small o. Loosely speaking, for
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Scene Planes & Homographies Lecture 19 March 24, 2005 2 In our last lecture, we examined various
More informationJordan Canonical Form
Jordan Canonical Form Massoud Malek Jordan normal form or Jordan canonical form (named in honor of Camille Jordan) shows that by changing the basis, a given square matrix M can be transformed into a certain
More informationSingular 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 information1 Linearity and Linear Systems
Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 78 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)
More informationB553 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 informationLecture 2: VectorVector Operations
Lecture 2: VectorVector Operations VectorVector Operations Addition of two vectors Geometric representation of addition and subtraction of vectors Vectors and points Dot product of two vectors Geometric
More informationL3: Review of linear algebra and MATLAB
L3: Review of linear algebra and MATLAB Vector and matrix notation Vectors Matrices Vector spaces Linear transformations Eigenvalues and eigenvectors MATLAB primer CSCE 666 Pattern Analysis Ricardo GutierrezOsuna
More informationTopics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij
Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix
More informationBasic Concepts in Matrix Algebra
Basic Concepts in Matrix Algebra An column array of p elements is called a vector of dimension p and is written as x p 1 = x 1 x 2. x p. The transpose of the column vector x p 1 is row vector x = [x 1
More informationPrincipal Component Analysis (PCA)
Principal Component Analysis (PCA) Salvador Dalí, Galatea of the Spheres CSC411/2515: Machine Learning and Data Mining, Winter 2018 Michael Guerzhoy and Lisa Zhang Some slides from Derek Hoiem and Alysha
More informationThe Cartan Dieudonné Theorem
Chapter 7 The Cartan Dieudonné Theorem 7.1 Orthogonal Reflections Orthogonal symmetries are a very important example of isometries. First let us review the definition of a (linear) projection. Given a
More informationStudy Notes on Matrices & Determinants for GATE 2017
Study Notes on Matrices & Determinants for GATE 2017 Matrices and Determinates are undoubtedly one of the most scoring and high yielding topics in GATE. At least 34 questions are always anticipated from
More informationSolutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015
Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See
More informationChapter 8. Rigid transformations
Chapter 8. Rigid transformations We are about to start drawing figures in 3D. There are no builtin routines for this purpose in PostScript, and we shall have to start more or less from scratch in extending
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 II Homework #1 solutions A = This means that every eigenvector with eigenvalue λ = 1 must have the form
Linear algebra II Homework # solutions. Find the eigenvalues and the eigenvectors of the matrix 4 6 A =. 5 Since tra = 9 and deta = = 8, the characteristic polynomial is f(λ) = λ (tra)λ+deta = λ 9λ+8 =
More informationMATH 369 Linear Algebra
Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine
More informationFinal Exam. Linear Algebra Summer 2011 Math S2010X (3) Corrin Clarkson. August 10th, Solutions
Final Exam Linear Algebra Summer Math SX (3) Corrin Clarkson August th, Name: Solutions Instructions: This is a closed book exam. You may not use the textbook, notes or a calculator. You will have 9 minutes
More informationLinear 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 informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
More informationCOMP6237 Data Mining Covariance, EVD, PCA & SVD. Jonathon Hare
COMP6237 Data Mining Covariance, EVD, PCA & SVD Jonathon Hare jsh2@ecs.soton.ac.uk Variance and Covariance Random Variables and Expected Values Mathematicians talk variance (and covariance) in terms of
More informationExamples and MatLab. Vector and Matrix Material. Matrix Addition R = A + B. Matrix Equality A = B. Matrix Multiplication R = A * B.
Vector and Matrix Material Examples and MatLab Matrix = Rectangular array of numbers, complex numbers or functions If r rows & c columns, r*c elements, r*c is order of matrix. A is n * m matrix Square
More informationLinear Algebra Final Exam Review
Linear Algebra Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More informationSpring, 2012 CIS 515. Fundamentals of Linear Algebra and Optimization Jean Gallier
Spring 0 CIS 55 Fundamentals of Linear Algebra and Optimization Jean Gallier Homework 5 & 6 + Project 3 & 4 Note: Problems B and B6 are for extra credit April 7 0; Due May 7 0 Problem B (0 pts) Let A be
More informationLECTURE VI: SELFADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY
LECTURE VI: SELFADJOINT AND UNITARY OPERATORS MAT 204  FALL 2006 PRINCETON UNIVERSITY ALFONSO SORRENTINO 1 Adjoint of a linear operator Note: In these notes, V will denote a ndimensional euclidean vector
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 informationDot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.
Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................
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 informationMLCC 2015 Dimensionality Reduction and PCA
MLCC 2015 Dimensionality Reduction and PCA Lorenzo Rosasco UNIGEMITIIT June 25, 2015 Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component
More informationCS168: The Modern Algorithmic Toolbox Lecture #8: How PCA Works
CS68: The Modern Algorithmic Toolbox Lecture #8: How PCA Works Tim Roughgarden & Gregory Valiant April 20, 206 Introduction Last lecture introduced the idea of principal components analysis (PCA). The
More information1 Planar rotations. Math Abstract Linear Algebra Fall 2011, section E1 Orthogonal matrices and rotations
Math 46  Abstract Linear Algebra Fall, section E Orthogonal matrices and rotations Planar rotations Definition: A planar rotation in R n is a linear map R: R n R n such that there is a plane P R n (through
More informationExercise Sheet 1. 1 Probability revision 1: Studentt as an infinite mixture of Gaussians
Exercise Sheet 1 1 Probability revision 1: Studentt as an infinite mixture of Gaussians Show that an infinite mixture of Gaussian distributions, with Gamma distributions as mixing weights in the following
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 informationIntro Vectors 2D implicit curves 2D parametric curves. Graphics 2011/2012, 4th quarter. Lecture 2: vectors, curves, and surfaces
Lecture 2, curves, and surfaces Organizational remarks Tutorials: Tutorial 1 will be online later today TA sessions for questions start next week Practicals: Exams: Make sure to find a team partner very
More informationDefinition (T invariant subspace) Example. Example
Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin
More informationMath 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 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 informationSection 13.4 The Cross Product
Section 13.4 The Cross Product Multiplying Vectors 2 In this section we consider the more technical multiplication which can be defined on vectors in 3space (but not vectors in 2space). 1. Basic Definitions
More informationTopic 2 Quiz 2. choice C implies B and B implies C. correctchoice C implies B, but B does not imply C
Topic 1 Quiz 1 text A reduced rowechelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correctchoice may have 0, 1, 2, or 3 choice may have 0,
More informationPrinciple Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA
Principle Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA Principle Components Analysis: Uses one group of variables (we will call this X) In
More informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More information