DOCSLIB.ORG

Linear Algebra As an Introduction to Abstract Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Algebra As an Introduction to Abstract Mathematics Linear Algebra As an Introduction to Abstract Mathematics Lecture Notes for MAT67 University of California, Davis written Fall 2007, last updated November 15, 2016 Isaiah Lankham Bruno Nachtergaele Anne Schilling Copyright c 2007 by the authors. These lecture notes may be reproduced in their entirety for non-commercial purposes. Contents 1 What is Linear Algebra? 1 1.1 Introduction . .1 1.2 What is Linear Algebra? . .2 1.3 Systems of linear equations . .4 1.3.1 Linear equations . .4 1.3.2 Non-linear equations . .5 1.3.3 Linear transformations . .5 1.3.4 Applications of linear equations . .7 Exercises . .9 2 Introduction to Complex Numbers 11 2.1 Definition of complex numbers . 11 2.2 Operations on complex numbers . 12 2.2.1 Addition and subtraction of complex numbers . 12 2.2.2 Multiplication and division of complex numbers . 13 2.2.3 Complex conjugation . 15 2.2.4 The modulus (a.k.a. norm, length, or magnitude) . 16 2.2.5 Complex numbers as vectors in R2 ................... 18 2.3 Polar form and geometric interpretation for C ................. 19 2.3.1 Polar form for complex numbers . 19 2.3.2 Geometric multiplication for complex numbers . 20 2.3.3 Exponentiation and root extraction . 21 2.3.4 Some complex elementary functions . 22 Exercises . 24 ii 3 The Fundamental Theorem of Algebra and Factoring Polynomials 26 3.1 The Fundamental Theorem of Algebra . 26 3.2 Factoring polynomials . 30 Exercises . 34 4 Vector Spaces 36 4.1 Definition of vector spaces . 36 4.2 Elementary properties of vector spaces . 39 4.3 Subspaces . 40 4.4 Sums and direct sums . 42 Exercises . 46 5 Span and Bases 48 5.1 Linear span . 48 5.2 Linear independence . 50 5.3 Bases . 55 5.4 Dimension . 57 Exercises . 61 6 Linear Maps 64 6.1 Definition and elementary properties . 64 6.2 Null spaces . 67 6.3 Range . 69 6.4 Homomorphisms . 70 6.5 The dimension formula . 71 6.6 The matrix of a linear map . 73 6.7 Invertibility . 78 Exercises . 82 7 Eigenvalues and Eigenvectors 85 7.1 Invariant subspaces . 85 7.2 Eigenvalues . 86 7.3 Diagonal matrices . 89 7.4 Existence of eigenvalues . 90 iii 7.5 Upper triangular matrices . 91 7.6 Diagonalization of 2 × 2 matrices and applications . 96 Exercises . 98 8 Permutations and the Determinant of a Square Matrix 102 8.1 Permutations . 102 8.1.1 Definition of permutations . 102 8.1.2 Composition of permutations . 105 8.1.3 Inversions and the sign of a permutation . 107 8.2 Determinants . 110 8.2.1 Summations indexed by the set of all permutations . 110 8.2.2 Properties of the determinant . 112 8.2.3 Further properties and applications . 115 8.2.4 Computing determinants with cofactor expansions . 116 Exercises . 118 9 Inner Product Spaces 120 9.1 Inner product . 120 9.2 Norms . 122 9.3 Orthogonality . 124 9.4 Orthonormal bases . 127 9.5 The Gram-Schmidt orthogonalization procedure . 129 9.6 Orthogonal projections and minimization problems . 132 Exercises . 136 10 Change of Bases 139 10.1 Coordinate vectors . 139 10.2 Change of basis transformation . 141 Exercises . 145 11 The Spectral Theorem for Normal Linear Maps 147 11.1 Self-adjoint or hermitian operators . 147 11.2 Normal operators . 149 11.3 Normal operators and the spectral decomposition . 151 iv 11.4 Applications of the Spectral Theorem: diagonalization . 153 11.5 Positive operators . 157 11.6 Polar decomposition . 158 11.7 Singular-value decomposition . 159 Exercises . 161 List of Appendices A Supplementary Notes on Matrices and Linear Systems 164 A.1 From linear systems to matrix equations . 164 A.1.1 Definition of and notation for matrices . 165 A.1.2 Using matrices to encode linear systems . 168 A.2 Matrix arithmetic . 171 A.2.1 Addition and scalar multiplication . 171 A.2.2 Multiplication of matrices . 175 A.2.3 Invertibility of square matrices . 179 A.3 Solving linear systems by factoring the coefficient matrix . 181 A.3.1 Factorizing matrices using Gaussian elimination . 182 A.3.2 Solving homogeneous linear systems . 192 A.3.3 Solving inhomogeneous linear systems . 195 A.3.4 Solving linear systems with LU-factorization . 199 A.4 Matrices and linear maps . 204 A.4.1 The canonical matrix of a linear map . 204 A.4.2 Using linear maps to solve linear systems . 205 A.5 Special operations on matrices . 211 A.5.1 Transpose and conjugate transpose . 211 A.5.2 The trace of a square matrix . 212 Exercises . 214 B The Language of Sets and Functions 218 B.1 Sets . 218 B.2 Subset, union, intersection, and Cartesian product . 220 B.3 Relations . 222 v B.4 Functions . 223 C Summary of Algebraic Structures Encountered 226 C.1 Binary operations and scaling operations . 226 C.2 Groups, fields, and vector spaces . 229 C.3 Rings and algebras . 233 D Some Common Math Symbols and Abbreviations 236 E Summary of Notation Used 243 F Movie Scripts 246 vi Chapter 1 What is Linear Algebra? 1.1 Introduction This book aims to bridge the gap between the mainly computation-oriented lower division undergraduate classes and the abstract mathematics encountered in more advanced mathe- matics courses. The goal of this book is threefold: 1. You will learn Linear Algebra, which is one of the most widely used mathematical theories around. Linear Algebra finds applications in virtually every area of mathe- matics, including multivariate calculus, differential equations, and probability theory. It is also widely applied in fields like physics, chemistry, economics, psychology, and engineering. You are even relying on methods from Linear Algebra every time you use an internet search like Google, the Global Positioning System (GPS), or a cellphone. 2. You will acquire computational skills to solve linear systems of equations, perform operations on matrices, calculate eigenvalues, and find determinants of matrices. 3. In the setting of Linear Algebra, you will be introduced to abstraction. As the theory.
Load more

Recommended publications
  • HILBERT SPACE GEOMETRY Definition: a Vector Space Over Is a Set V (Whose Elements Are Called Vectors) Together with a Binary Operation
    HILBERT SPACE GEOMETRY Definition: A vector space over is a set V (whose elements are called vectors) together with a binary operation +:V×V→V, which is called vector addition, and an external binary operation ⋅: ×V→V, which is called scalar multiplication, such that (i) (V,+) is a commutative group (whose neutral element is called zero vector) and (ii) for all λ,µ∈ , x,y∈V: λ(µx)=(λµ)x, 1 x=x, λ(x+y)=(λx)+(λy), (λ+µ)x=(λx)+(µx), where the image of (x,y)∈V×V under + is written as x+y and the image of (λ,x)∈ ×V under ⋅ is written as λx or as λ⋅x. Exercise: Show that the set 2 together with vector addition and scalar multiplication defined by x y x + y 1 + 1 = 1 1 + x 2 y2 x 2 y2 x λx and λ 1 = 1 , λ x 2 x 2 respectively, is a vector space. 1 Remark: Usually we do not distinguish strictly between a vector space (V,+,⋅) and the set of its vectors V. For example, in the next definition V will first denote the vector space and then the set of its vectors. Definition: If V is a vector space and M⊆V, then the set of all linear combinations of elements of M is called linear hull or linear span of M. It is denoted by span(M). By convention, span(∅)={0}. Proposition: If V is a vector space, then the linear hull of any subset M of V (together with the restriction of the vector addition to M×M and the restriction of the scalar multiplication to ×M) is also a vector space.
    [Show full text]
  • LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
    LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
    [Show full text]
  • Things You Need to Know About Linear Algebra Math 131 Multivariate
    the standard (x; y; z) coordinate three-space. Nota- tions for vectors. Geometric interpretation as vec- tors as displacements as well as vectors as points. Vector addition, the zero vector 0, vector subtrac- Things you need to know about tion, scalar multiplication, their properties, and linear algebra their geometric interpretation. Math 131 Multivariate Calculus We start using these concepts right away. In D Joyce, Spring 2014 chapter 2, we'll begin our study of vector-valued functions, and we'll use the coordinates, vector no- The relation between linear algebra and mul- tation, and the rest of the topics reviewed in this tivariate calculus. We'll spend the first few section. meetings reviewing linear algebra. We'll look at just about everything in chapter 1. Section 1.2. Vectors and equations in R3. Stan- Linear algebra is the study of linear trans- dard basis vectors i; j; k in R3. Parametric equa- formations (also called linear functions) from n- tions for lines. Symmetric form equations for a line dimensional space to m-dimensional space, where in R3. Parametric equations for curves x : R ! R2 m is usually equal to n, and that's often 2 or 3. in the plane, where x(t) = (x(t); y(t)). A linear transformation f : Rn ! Rm is one that We'll use standard basis vectors throughout the preserves addition and scalar multiplication, that course, beginning with chapter 2. We'll study tan- is, f(a + b) = f(a) + f(b), and f(ca) = cf(a). We'll gent lines of curves and tangent planes of surfaces generally use bold face for vectors and for vector- in that chapter and throughout the course.
    [Show full text]
  • Schaum's Outline of Linear Algebra (4Th Edition)
    SCHAUM’S SCHAUM’S outlines outlines Linear Algebra Fourth Edition Seymour Lipschutz, Ph.D. Temple University Marc Lars Lipson, Ph.D. University of Virginia Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009, 2001, 1991, 1968 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior writ- ten permission of the publisher. ISBN: 978-0-07-154353-8 MHID: 0-07-154353-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154352-1, MHID: 0-07-154352-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected]. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work.
    [Show full text]
  • Does Geometric Algebra Provide a Loophole to Bell's Theorem?
    Discussion Does Geometric Algebra provide a loophole to Bell’s Theorem? Richard David Gill 1 1 Leiden University, Faculty of Science, Mathematical Institute; [email protected] Version October 30, 2019 submitted to Entropy Abstract: Geometric Algebra, championed by David Hestenes as a universal language for physics, was used as a framework for the quantum mechanics of interacting qubits by Chris Doran, Anthony Lasenby and others. Independently of this, Joy Christian in 2007 claimed to have refuted Bell’s theorem with a local realistic model of the singlet correlations by taking account of the geometry of space as expressed through Geometric Algebra. A series of papers culminated in a book Christian (2014). The present paper first explores Geometric Algebra as a tool for quantum information and explains why it did not live up to its early promise. In summary, whereas the mapping between 3D geometry and the mathematics of one qubit is already familiar, Doran and Lasenby’s ingenious extension to a system of entangled qubits does not yield new insight but just reproduces standard QI computations in a clumsy way. The tensor product of two Clifford algebras is not a Clifford algebra. The dimension is too large, an ad hoc fix is needed, several are possible. I further analyse two of Christian’s earliest, shortest, least technical, and most accessible works (Christian 2007, 2011), exposing conceptual and algebraic errors. Since 2015, when the first version of this paper was posted to arXiv, Christian has published ambitious extensions of his theory in RSOS (Royal Society - Open Source), arXiv:1806.02392, and in IEEE Access, arXiv:1405.2355.
    [Show full text]
  • Determinants Math 122 Calculus III D Joyce, Fall 2012
    Determinants Math 122 Calculus III D Joyce, Fall 2012 What they are. A determinant is a value associated to a square array of numbers, that square array being called a square matrix. For example, here are determinants of a general 2 × 2 matrix and a general 3 × 3 matrix. a b = ad − bc: c d a b c d e f = aei + bfg + cdh − ceg − afh − bdi: g h i The determinant of a matrix A is usually denoted jAj or det (A). You can think of the rows of the determinant as being vectors. For the 3×3 matrix above, the vectors are u = (a; b; c), v = (d; e; f), and w = (g; h; i). Then the determinant is a value associated to n vectors in Rn. There's a general definition for n×n determinants. It's a particular signed sum of products of n entries in the matrix where each product is of one entry in each row and column. The two ways you can choose one entry in each row and column of the 2 × 2 matrix give you the two products ad and bc. There are six ways of chosing one entry in each row and column in a 3 × 3 matrix, and generally, there are n! ways in an n × n matrix. Thus, the determinant of a 4 × 4 matrix is the signed sum of 24, which is 4!, terms. In this general definition, half the terms are taken positively and half negatively. In class, we briefly saw how the signs are determined by permutations.
    [Show full text]
  • Problems in Abstract Algebra
    STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
    [Show full text]
  • Abstract Algebra
    Abstract Algebra Martin Isaacs, University of Wisconsin-Madison (Chair) Patrick Bahls, University of North Carolina, Asheville Thomas Judson, Stephen F. Austin State University Harriet Pollatsek, Mount Holyoke College Diana White, University of Colorado Denver 1 Introduction What follows is a report summarizing the proposals of a group charged with developing recommendations for undergraduate curricula in abstract algebra.1 We begin by articulating the principles that shaped the discussions that led to these recommendations. We then indicate several learning goals; some of these address specific content areas and others address students' general development. Next, we include three sample syllabi, each tailored to meet the needs of specific types of institutions and students. Finally, we present a brief list of references including sample texts. 2 Guiding Principles We lay out here several principles that underlie our recommendations for undergraduate Abstract Algebra courses. Although these principles are very general, we indicate some of their specific implications in the discussions of learning goals and curricula below. Diversity of students We believe that a course in Abstract Algebra is valuable for a wide variety of students, including mathematics majors, mathematics education majors, mathematics minors, and majors in STEM disciplines such as physics, chemistry, and computer science. Such a course is essential preparation for secondary teaching and for many doctoral programs in mathematics. Moreover, algebra can capture the imagination of students whose attraction to mathematics is primarily to structure and abstraction (for example, 1As with any document that is produced by a committee, there were some disagreements and compromises. The committee members had many lively and spirited communications on what undergraduate Abstract Algebra should look like for the next ten years.
    [Show full text]
  • Abstract Algebra
    Abstract Algebra Paul Melvin Bryn Mawr College Fall 2011 lecture notes loosely based on Dummit and Foote's text Abstract Algebra (3rd ed) Prerequisite: Linear Algebra (203) 1 Introduction Pure Mathematics Algebra Analysis Foundations (set theory/logic) G eometry & Topology What is Algebra? • Number systems N = f1; 2; 3;::: g \natural numbers" Z = f:::; −1; 0; 1; 2;::: g \integers" Q = ffractionsg \rational numbers" R = fdecimalsg = pts on the line \real numbers" p C = fa + bi j a; b 2 R; i = −1g = pts in the plane \complex nos" k polar form re iθ, where a = r cos θ; b = r sin θ a + bi b r θ a p Note N ⊂ Z ⊂ Q ⊂ R ⊂ C (all proper inclusions, e.g. 2 62 Q; exercise) There are many other important number systems inside C. 2 • Structure \binary operations" + and · associative, commutative, and distributive properties \identity elements" 0 and 1 for + and · resp. 2 solve equations, e.g. 1 ax + bx + c = 0 has two (complex) solutions i p −b ± b2 − 4ac x = 2a 2 2 2 2 x + y = z has infinitely many solutions, even in N (thei \Pythagorian triples": (3,4,5), (5,12,13), . ). n n n 3 x + y = z has no solutions x; y; z 2 N for any fixed n ≥ 3 (Fermat'si Last Theorem, proved in 1995 by Andrew Wiles; we'll give a proof for n = 3 at end of semester). • Abstract systems groups, rings, fields, vector spaces, modules, . A group is a set G with an associative binary operation ∗ which has an identity element e (x ∗ e = x = e ∗ x for all x 2 G) and inverses for each of its elements (8 x 2 G; 9 y 2 G such that x ∗ y = y ∗ x = e).
    [Show full text]
  • Spanning Sets the Only Algebraic Operations That Are Defined in a Vector Space V Are Those of Addition and Scalar Multiplication
    i i “main” 2007/2/16 page 258 i i 258 CHAPTER 4 Vector Spaces 23. Show that the set of all solutions to the nonhomoge- and let neous differential equation S1 + S2 = {v ∈ V : y + a1y + a2y = F(x), v = x + y for some x ∈ S1 and y ∈ S2} . where F(x) is nonzero on an interval I, is not a sub- 2 space of C (I). (a) Show that, in general, S1 ∪ S2 is not a subspace of V . 24. Let S1 and S2 be subspaces of a vector space V . Let (b) Show that S1 ∩ S2 is a subspace of V . S ∪ S ={v ∈ V : v ∈ S or v ∈ S }, 1 2 1 2 (c) Show that S1 + S2 is a subspace of V . S1 ∩ S2 ={v ∈ V : v ∈ S1 and v ∈ S2}, 4.4 Spanning Sets The only algebraic operations that are defined in a vector space V are those of addition and scalar multiplication. Consequently, the most general way in which we can combine the vectors v1, v2,...,vk in V is c1v1 + c2v2 +···+ckvk, (4.4.1) where c1,c2,...,ck are scalars. An expression of the form (4.4.1) is called a linear combination of v1, v2,...,vk. Since V is closed under addition and scalar multiplica- tion, it follows that the foregoing linear combination is itself a vector in V . One of the questions we wish to answer is whether every vector in a vector space can be obtained by taking linear combinations of a finite set of vectors. The following terminology is used in the case when the answer to this question is affirmative: DEFINITION 4.4.1 If every vector in a vector space V can be written as a linear combination of v1, v2, ..., vk, we say that V is spanned or generated by v1, v2, ..., vk and call the set of vectors {v1, v2,...,vk} a spanning set for V .
    [Show full text]
  • 1 Algebra Vs. Abstract Algebra 2 Abstract Number Systems in Linear
    MATH 2135, LINEAR ALGEBRA, Winter 2017 and multiplication is also called the “logical and” operation. For example, we Handout 1: Lecture Notes on Fields can calculate like this: Peter Selinger 1 · ((1 + 0) + 1) + 1 = 1 · (1 + 1) + 1 = 1 · 0 + 1 = 0+1 1 Algebra vs. abstract algebra = 1. Operations such as addition and multiplication can be considered at several dif- 2 Abstract number systems in linear algebra ferent levels: • Arithmetic deals with specific calculation rules, such as 8 + 3 = 11. It is As you already know, Linear Algebra deals with subjects such as matrix multi- usually taught in elementary school. plication, linear combinations, solutions of systems of linear equations, and so on. It makes heavy use of addition, subtraction, multiplication, and division of • Algebra deals with the idea that operations satisfy laws, such as a(b+c)= scalars (think, for example, of the rule for multiplying matrices). ab + ac. Such laws can be used, among other things, to solve equations It turns out that most of what we do in linear algebra does not rely on the spe- such as 3x + 5 = 14. cific laws of arithmetic. Linear algebra works equally well over “alternative” • Abstract algebra is the idea that we can use the laws of algebra, such as arithmetics. a(b + c) = ab + ac, while abandoning the rules of arithmetic, such as Example 2.1. Consider multiplying two matrices, using arithmetic modulo 2 8 + 3 = 11. Thus, in abstract algebra, we are able to speak of entirely instead of the usual arithmetic. different “number” systems, for example, systems in which 1+1=0.
    [Show full text]
  • Span, Linear Independence and Basis Rank and Nullity
    Remarks for Exam 2 in Linear Algebra Span, linear independence and basis The span of a set of vectors is the set of all linear combinations of the vectors. A set of vectors is linearly independent if the only solution to c1v1 + ::: + ckvk = 0 is ci = 0 for all i. Given a set of vectors, you can determine if they are linearly independent by writing the vectors as the columns of the matrix A, and solving Ax = 0. If there are any non-zero solutions, then the vectors are linearly dependent. If the only solution is x = 0, then they are linearly independent. A basis for a subspace S of Rn is a set of vectors that spans S and is linearly independent. There are many bases, but every basis must have exactly k = dim(S) vectors. A spanning set in S must contain at least k vectors, and a linearly independent set in S can contain at most k vectors. A spanning set in S with exactly k vectors is a basis. A linearly independent set in S with exactly k vectors is a basis. Rank and nullity The span of the rows of matrix A is the row space of A. The span of the columns of A is the column space C(A). The row and column spaces always have the same dimension, called the rank of A. Let r = rank(A). Then r is the maximal number of linearly independent row vectors, and the maximal number of linearly independent column vectors. So if r < n then the columns are linearly dependent; if r < m then the rows are linearly dependent.
    [Show full text]