The Second Undergraduate Level Course in Linear Algebra
|
|
|
- Matthew Myron Conley
- 7 years ago
- Views:
Transcription
1 The Second Undergraduate Level Course in Linear Algebra University of Massachusetts Dartmouth Joint Mathematics Meetings New Orleans, January 7, 11
2 Outline of Talk Linear Algebra Curriculum Study Group (LACSG) Recommendations LACSG recommended first course topics Core topics for a second course Numerical Linear Algebra Undergraduate explorations and research in linear algebra
3 Linear Algebra Curriculum Study Group Organizers: David Carlson, Charles R. Johnson, David C. Lay, and Duane Porter The William and Mary Workshop Panel: Glenn Adamson, Paul Bengtson, James Bunch, David Carlson, Jane Day, Guershon Harel, Roger Horn
4 LACSG Recommendations 1. The syllabus must respond to the client disciplines.
5 LACSG Recommendations 1. The syllabus must respond to the client disciplines. 2. The first course should be matrix oriented.
6 LACSG Recommendations 1. The syllabus must respond to the client disciplines. 2. The first course should be matrix oriented. 3. Faculty should consider the needs and interests of students.
7 LACSG Recommendations 1. The syllabus must respond to the client disciplines. 2. The first course should be matrix oriented. 3. Faculty should consider the needs and interests of students. 4. Faculty should use technology.
8 LACSG Recommendations 1. The syllabus must respond to the client disciplines. 2. The first course should be matrix oriented. 3. Faculty should consider the needs and interests of students. 4. Faculty should use technology. 5. At least one follow-up course should be required.
9 The LACSG Core Course Topics Core topics 1. Matrix Addition and Multiplication 2. Systems of Linear Equations 3. Determinants 4. Properties of R n 5. Eigenvalues and Eigenvectors 6. More on Orthogonality 7. Supplementary Topics
10 1. Matrix Addition and Multiplication matrix addition scalar multiplication matrix multiplication transposition associativity of matrix multiplication operations with partitioned matrices
11 2. Systems of Linear Equations Gaussian elimination elementary matrices echelon and reduced row echelon form existence and uniqueness of solutions matrix inverses row reduction interpreted as LU-factorization
12 3. Determinants cofactor expansion determinants and row operations det AB = det A det B Cramer s Rule
13 4. Properties of R n linear combinations: linear dependence and independence bases of R n subspaces of R n : spanning set, basis, dimension, row space and column space, null space matrices of linear transforms rank: row rank=column rank, products, connections with invertible submatrices systems of equations revisited: solution theory, rank+nullity=number of columns inner product: length and orthogonality, orthogonal /orthonormal sets and bases, orthogonal matrices
14 5. Eigenvalues and Eigenvectors the equation Ax = λx characteristic polynomial and identification of some of its coefficients (trace and determinant), algebraic multiplicity of eigenvalues eigenspaces, geometric multiplicity similarity: distinct eigenvalues and diagonalization symmetric matrices: orthogonal diagonalization, quadratic forms
15 6. More on Orthogonality orthogonal projection onto a subspace: Gram-Schmidt orthogonalization and interpretation as a QR factorization the least square solutions of inconsistent linear systems, with applications to data-fitting
16 7. Supplementary Topics computational experience abstract vector spaces linear transforms positive definite matrices reduction of a symmetric matrix by congruence singular value decomposition matrix norms
17 LACSG Recommendations Revisited 1. The syllabus must respond to the client disciplines. 2. The first course should be matrix oriented. 3. Faculty should consider the needs and interests of students. 4. Faculty should use technology. 5. At least one follow-up course should be required. There has been considerable progress with regards to the first four LACSG recommendations. Evidence of this comes from comparing the textbooks used today with those of years ago.
18 Challenges for the Future Challenge to cover the entire LACSG core curriculum in a single one semester course. At some universities this challenge has been increasing in recent years. The Supplementary Topics (Unit 7) are generally deferred to a second course. At many universities second courses are offered but not required.
19 Core topics for the second course Abstract vector spaces: function spaces and inner products Special classes of matrices: orthogonal and unitary, Hermitian, normal, positive definite. Singular value decomposition and other matrix factorizations Matrix norms The matrix exponential
20 A Numerical Linear Algebra Course Matrix norms and condition numbers Numerical algorithms for linear systems, least squares problems, and eigenvalue problems Numerical rank and the singular value decomposition Householder transformations Modified Gram-Schmidt process Sparse matrices and Krylov subspaces
21 Projects oriented courses Students work in teams on exploration/research projects Student develop MATLAB code for their projects Examples of projects Computer animation Applications to coordinate metrology Digital imaging applications
22 Computer animation Contractions, dilations, rotations, and translations are carried as matrix multiplications. Homogeneous coordinate system x a x = x 2, A = 0 1 b , Ax = x 1 + a x 2 + b 1
23 Computer animation Project make stick figure walk across screen press any key to start Design video game Meteor
24 Meteor video game >> help meteor Key(s) Left arrow or j Up arrow or k Right arrow or l Down arrow or ; Spacebar f h p Escape Action Rotate ship counterclockwise 15 degrees Thrust forward Rotate ship clockwise 15 degrees Apply air brakes Fire shot Flip ship (Turn ship 1 degrees) Send ship to the home (start) position Pause the game Quit the game
25 Meteor game description Each meteoroid is worth points. The speed and direction of each meteoroid are randomly chosen. As levels are completed, the nominal meteoroid speed increases proportionally. After level 2, the deploy time (how long before a new meteoroid is visible) and the number of meteoroids per level are held constant.
26 Coordinate Metrology Problems Write MATLAB program to fit a circle to data points Find the line that gives the best geometric (or orthogonal) fit to data points Find the best rectangular fit to a set of points. Other types of geometric fits
27 Output from Circlefit MATLAB function center (2.0432,1.0596), radius
28 Finding the line that gives best geometric fit to m data points Equation of line: c 1 + c 2 x + c 3 y = 0 with c2 2 + c2 3 = 1 Let r i = c 1 + c 2 x i + c 3 y i i = 1... m Find c 1, c 2, c 3 to minimize r 2 subject to c2 2 + c2 3 = 1. If A = [e x y] then r = Ac
29 Solution using QR and SVD factorizations If A = QR then where r 2 = Ac 2 = QRc 2 = Rc 2 = R 1 c 2 R 1 c = r 11 c 1 + r 12 c 2 + r 13 c 3 r 22 c 2 + r 23 c 3 r 33 c 3 To minimize R 1 c 2 subject to c2 2 + c2 3 = 1, let UΣV T be the singular value decomposition of [ ] r22 r R 2 = 23, r 33 set (c 2, c 3 ) = v T 2 and then solve for c 1.
30 Fitting Parallel and Perpendicular lines Given two data sets (x i, y i ), i = 1,..., m and (x i, y i ), i = m + 1,..., m + k Parallel lines: Set r i = c 1 + 0c 2 + c 3 x i + c 4 y i i = 1,..., m r i = 0c 1 + c 2 + c 3 x i + c 4 y i i = m + 1,..., m + k Perpendicular lines: Set r i = c 1 + 0c 2 + c 3 x i + c 4 y i i = 1,..., m r i = 0c 1 + c 2 + c 3 y i c 4 x i i = m + 1,..., m + k Solve for c 1, c 2, c 3, c 4 using QR and SVD factorizations.
31 Fitting rectangles Given 4 data sets each with m points. Set r i = c 1 + 0c 2 + 0c 3 + 0c 4 + c 5 x i + c 6 y i i = 1,..., m r i = 0c 1 + c 2 + 0c 3 + 0c 4 + c 5 y i c 6 x i i = m + 1,..., 2m r i = 0c 1 + 0c 2 + c 3 + 0c 4 + c 5 x i + c 6 y i i = 2m + 1,..., 3m r i = 0c 1 + 0c 2 + 0c 3 + c 4 + c 5 y i c 6 x i i = 3m + 1,..., 4m Solve for c 1, c 2, c 3, c 4, c 5, c 6 using QR and SVD factorizations. Determine the 4 points where the perpendicular lines intersect.
32 Plot of Rectangle Fit
33 SVD Image compression A black and white picture can be represented as a single m n matrix A with integer entries representing the grey levels of the pixels of the image. SVD compression of image (low rank approximation of A) Singular value decomposition: A = UΣV T, Σ = diag(σ 1,..., σ n ) and U T U = I m, V T V = I n Singular values: σ 1 σ 2... σ n 0 Best rank k approximation to A is σ 1 u 1 v T σ ku k v T k
34 7. SVD Compr. (1/128) abs(original approximated) 10 6 Singular Values (Log Scale)
35 8. SVD Compr. (3/128) abs(original approximated) 10 6 Singular Values (Log Scale)
36 9. SVD Compr. (5/128) abs(original approximated) 10 6 Singular Values (Log Scale)
37 10. SVD Compr. (7/128) abs(original approximated) 10 6 Singular Values (Log Scale)
38 11. SVD Compr. (9/128) abs(original approximated) 10 6 Singular Values (Log Scale)
39 12. SVD Compr. (13/128) abs(original approximated) 10 6 Singular Values (Log Scale)
40 13. SVD Compr. (17/128) abs(original approximated) 10 6 Singular Values (Log Scale)
41 14. SVD Compr. (21/128) abs(original approximated) 10 6 Singular Values (Log Scale)
42 15. SVD Compr. (25/128) abs(original approximated) 10 6 Singular Values (Log Scale)
43 SVD Image compression Color Images A color image can be represented using three m n matrices A 1, A 2, A 3. Project: Compare 2 approaches to SVD compression Method 1: Do svd compression to all 3 matrices Method 2: Do svd compression on a single 3m n block matrix. W = A 1 A 2 A 3 Compare quality of images and computational efficiency of methods
44 Conclusions One undergraduate course in linear algebra is not enough. Many important topics get left out of the first course. A second course is necessary to reinforce material from the first course. The first course ends just before one gets to the most interesting and useful topics. Linear algebra is an ideal subject for undergraduate research and explorations.
Linear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
Applied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
Chapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated
SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT-240 Action Taken (Please Check One) New Course Initiated
LINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every
by the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
Lecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
Recall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
Similarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
Mean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.
Orthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
Similar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
CS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
Inner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
Nonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., Richard-Plouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
LINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
MAT 242 Test 3 SOLUTIONS, FORM A
MAT Test SOLUTIONS, FORM A. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. A = 8 a. [5 points] Find the orthogonal projection
[1] Diagonal factorization
![[1] Diagonal factorization](/thumbs/40/21671034.jpg "[1] Diagonal factorization")
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
MAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms
3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
Eigenvalues and Eigenvectors
Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution
Lecture notes on linear algebra
Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
Numerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,
Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
Section 6.1 - Inner Products and Norms
Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators
is in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
Name: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
Examination paper for TMA4115 Matematikk 3
Department of Mathematical Sciences Examination paper for TMA45 Matematikk 3 Academic contact during examination: Antoine Julien a, Alexander Schmeding b, Gereon Quick c Phone: a 73 59 77 82, b 40 53 99
( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column
Factorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
Orthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
MATH 551 - APPLIED MATRIX THEORY
MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3
Math 24 FINAL EXAM (2/9/9 - SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r
ISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
QUALITY ENGINEERING PROGRAM
QUALITY ENGINEERING PROGRAM Production engineering deals with the practical engineering problems that occur in manufacturing planning, manufacturing processes and in the integration of the facilities and
Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree of PhD of Engineering in Informatics
INTERNATIONAL BLACK SEA UNIVERSITY COMPUTER TECHNOLOGIES AND ENGINEERING FACULTY ELABORATION OF AN ALGORITHM OF DETECTING TESTS DIMENSIONALITY Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree
Numerical Methods I Eigenvalue Problems
Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)
Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
MAT 242 Test 2 SOLUTIONS, FORM T
MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these
APPM4720/5720: Fast algorithms for big data. Gunnar Martinsson The University of Colorado at Boulder
APPM4720/5720: Fast algorithms for big data Gunnar Martinsson The University of Colorado at Boulder Course objectives: The purpose of this course is to teach efficient algorithms for processing very large
5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal
University of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
Chapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
Review Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
Elementary Linear Algebra
Elementary Linear Algebra Kuttler January, Saylor URL: http://wwwsaylororg/courses/ma/ Saylor URL: http://wwwsaylororg/courses/ma/ Contents Some Prerequisite Topics Sets And Set Notation Functions Graphs
Matrices and Linear Algebra
Chapter 2 Matrices and Linear Algebra 2. Basics Definition 2... A matrix is an m n array of scalars from a given field F. The individual values in the matrix are called entries. Examples. 2 3 A = 2 4 2
Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal
ALGEBRAIC EIGENVALUE PROBLEM
ALGEBRAIC EIGENVALUE PROBLEM BY J. H. WILKINSON, M.A. (Cantab.), Sc.D. Technische Universes! Dsrmstedt FACHBEREICH (NFORMATiK BIBL1OTHEK Sachgebieto:. Standort: CLARENDON PRESS OXFORD 1965 Contents 1.
Linear Algebra: Determinants, Inverses, Rank
D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of
Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
![Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj](/thumbs/31/14930763.jpg "Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj")
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
Least-Squares Intersection of Lines
Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function
17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):
Modélisation et résolutions numérique et symbolique
Modélisation et résolutions numérique et symbolique via les logiciels Maple et Matlab Jeremy Berthomieu Mohab Safey El Din Stef Graillat Mohab.Safey@lip6.fr Outline Previous course: partial review of what
Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours
MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
Linearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
UNIVERSITY OF PUNE, PUNE 411007. BOARD OF STUDIES IN MATHEMATICS SYLLABUS
UNIVERSITY OF PUNE, PUNE 411007. BOARD OF STUDIES IN MATHEMATICS SYLLABUS F.Y.B.Sc (MATHEMATICS) PAPER 1 ALGEBRA AND GEOMETRY FIRST TERM 1) Sets (4 Lectures) 1.1 Power set of a set, Product of two sets.
The Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
The Geometry of Polynomial Division and Elimination
The Geometry of Polynomial Division and Elimination Kim Batselier, Philippe Dreesen Bart De Moor Katholieke Universiteit Leuven Department of Electrical Engineering ESAT/SCD/SISTA/SMC May 2012 1 / 26 Outline
More than you wanted to know about quadratic forms
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences More than you wanted to know about quadratic forms KC Border Contents 1 Quadratic forms 1 1.1 Quadratic forms on the unit
9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
Inner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
Lecture 1: Schur s Unitary Triangularization Theorem
Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections
4 MT210 Notebook 4 3. 4.1 Eigenvalues and Eigenvectors... 3. 4.1.1 Definitions; Graphical Illustrations... 3
MT Notebook Fall / prepared by Professor Jenny Baglivo c Copyright 9 by Jenny A. Baglivo. All Rights Reserved. Contents MT Notebook. Eigenvalues and Eigenvectors................................... Definitions;
18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2-106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 2-06 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are right-hand-sides b for which A x = b has no solution (a) What
AMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 19: SVD revisited; Software for Linear Algebra Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 9 Outline 1 Computing
Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
Numerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and Non-Square Systems
Numerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and Non-Square Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001,
u = [ 2 4 5] has one row with three components (a 3 v = [2 4 5] has three rows separated by semicolons (a 3 w = 2:5 generates the row vector w = [ 2 3
![u = [ 2 4 5] has one row with three components (a 3 v = [2 4 5] has three rows separated by semicolons (a 3 w = 2:5 generates the row vector w = [ 2 3](/thumbs/40/21151686.jpg "u = [ 2 4 5] has one row with three components (a 3 v = [2 4 5] has three rows separated by semicolons (a 3 w = 2:5 generates the row vector w = [ 2 3")
MATLAB Tutorial You need a small numb e r of basic commands to start using MATLAB. This short tutorial describes those fundamental commands. You need to create vectors and matrices, to change them, and
Introduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
Linear Algebraic Equations, SVD, and the Pseudo-Inverse
Linear Algebraic Equations, SVD, and the Pseudo-Inverse Philip N. Sabes October, 21 1 A Little Background 1.1 Singular values and matrix inversion For non-smmetric matrices, the eigenvalues and singular
Chapter 7. Lyapunov Exponents. 7.1 Maps
Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average
Solving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
Solutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
Math 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
Matrices and Linear Algebra
4 Matrices and Linear Algebra Matrices and Linear Algebra............ 4-2 Matrices in MATLAB................ 4-4 Addition and Subtraction............... 4-6 Vector Products and Transpose............. 4-7