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Linear Independence, Span, and Basis of a Set of Vectors What Is Linear Independence?
LECTURENOTES · purdue university MA 26500 Kyle Kloster Linear Algebra October 22, 2014 Linear Independence, Span, and Basis of a Set of Vectors What is linear independence? A set of vectors S = fv1; ··· ; vkg is linearly independent if none of the vectors vi can be written as a linear combination of the other vectors, i.e. vj = α1v1 + ··· + αkvk. Suppose the vector vj can be written as a linear combination of the other vectors, i.e. there exist scalars αi such that vj = α1v1 + ··· + αkvk holds. (This is equivalent to saying that the vectors v1; ··· ; vk are linearly dependent). We can subtract vj to move it over to the other side to get an expression 0 = α1v1 + ··· αkvk (where the term vj now appears on the right hand side. In other words, the condition that \the set of vectors S = fv1; ··· ; vkg is linearly dependent" is equivalent to the condition that there exists αi not all of which are zero such that 2 3 α1 6α27 0 = v v ··· v 6 7 : 1 2 k 6 . 7 4 . 5 αk More concisely, form the matrix V whose columns are the vectors vi. Then the set S of vectors vi is a linearly dependent set if there is a nonzero solution x such that V x = 0. This means that the condition that \the set of vectors S = fv1; ··· ; vkg is linearly independent" is equivalent to the condition that \the only solution x to the equation V x = 0 is the zero vector, i.e. x = 0. How do you determine if a set is lin. -
Discover Linear Algebra Incomplete Preliminary Draft
Discover Linear Algebra Incomplete Preliminary Draft Date: November 28, 2017 L´aszl´oBabai in collaboration with Noah Halford All rights reserved. Approved for instructional use only. Commercial distribution prohibited. c 2016 L´aszl´oBabai. Last updated: November 10, 2016 Preface TO BE WRITTEN. Babai: Discover Linear Algebra. ii This chapter last updated August 21, 2016 c 2016 L´aszl´oBabai. Contents Notation ix I Matrix Theory 1 Introduction to Part I 2 1 (F, R) Column Vectors 3 1.1 (F) Column vector basics . 3 1.1.1 The domain of scalars . 3 1.2 (F) Subspaces and span . 6 1.3 (F) Linear independence and the First Miracle of Linear Algebra . 8 1.4 (F) Dot product . 12 1.5 (R) Dot product over R ................................. 14 1.6 (F) Additional exercises . 14 2 (F) Matrices 15 2.1 Matrix basics . 15 2.2 Matrix multiplication . 18 2.3 Arithmetic of diagonal and triangular matrices . 22 2.4 Permutation Matrices . 24 2.5 Additional exercises . 26 3 (F) Matrix Rank 28 3.1 Column and row rank . 28 iii iv CONTENTS 3.2 Elementary operations and Gaussian elimination . 29 3.3 Invariance of column and row rank, the Second Miracle of Linear Algebra . 31 3.4 Matrix rank and invertibility . 33 3.5 Codimension (optional) . 34 3.6 Additional exercises . 35 4 (F) Theory of Systems of Linear Equations I: Qualitative Theory 38 4.1 Homogeneous systems of linear equations . 38 4.2 General systems of linear equations . 40 5 (F, R) Affine and Convex Combinations (optional) 42 5.1 (F) Affine combinations . -
Linear Independence, the Wronskian, and Variation of Parameters
LINEAR INDEPENDENCE, THE WRONSKIAN, AND VARIATION OF PARAMETERS JAMES KEESLING In this post we determine when a set of solutions of a linear differential equation are linearly independent. We first discuss the linear space of solutions for a homogeneous differential equation. 1. Homogeneous Linear Differential Equations We start with homogeneous linear nth-order ordinary differential equations with general coefficients. The form for the nth-order type of equation is the following. dnx dn−1x (1) a (t) + a (t) + ··· + a (t)x = 0 n dtn n−1 dtn−1 0 It is straightforward to solve such an equation if the functions ai(t) are all constants. However, for general functions as above, it may not be so easy. However, we do have a principle that is useful. Because the equation is linear and homogeneous, if we have a set of solutions fx1(t); : : : ; xn(t)g, then any linear combination of the solutions is also a solution. That is (2) x(t) = C1x1(t) + C2x2(t) + ··· + Cnxn(t) is also a solution for any choice of constants fC1;C2;:::;Cng. Now if the solutions fx1(t); : : : ; xn(t)g are linearly independent, then (2) is the general solution of the differential equation. We will explain why later. What does it mean for the functions, fx1(t); : : : ; xn(t)g, to be linearly independent? The simple straightforward answer is that (3) C1x1(t) + C2x2(t) + ··· + Cnxn(t) = 0 implies that C1 = 0, C2 = 0, ::: , and Cn = 0 where the Ci's are arbitrary constants. This is the definition, but it is not so easy to determine from it just when the condition holds to show that a given set of functions, fx1(t); x2(t); : : : ; xng, is linearly independent. -
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). -
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. -
Parity Alternating Permutations Starting with an Odd Integer
Parity alternating permutations starting with an odd integer Frether Getachew Kebede1a, Fanja Rakotondrajaob aDepartment of Mathematics, College of Natural and Computational Sciences, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia; e-mail: [email protected] bD´epartement de Math´ematiques et Informatique, BP 907 Universit´ed’Antananarivo, 101 Antananarivo, Madagascar; e-mail: [email protected] Abstract A Parity Alternating Permutation of the set [n] = 1, 2,...,n is a permutation with { } even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the numbers that count the PAPs with even as well as odd parity. We also study a subclass of PAPs being derangements as well, Parity Alternating Derangements (PADs). Moreover, by considering the parity of these PADs we look into their statistical property of excedance. Keywords: parity, parity alternating permutation, parity alternating derangement, excedance 2020 MSC: 05A05, 05A15, 05A19 arXiv:2101.09125v1 [math.CO] 22 Jan 2021 1. Introduction and preliminaries A permutation π is a bijection from the set [n]= 1, 2,...,n to itself and we will write it { } in standard representation as π = π(1) π(2) π(n), or as the product of disjoint cycles. ··· The parity of a permutation π is defined as the parity of the number of transpositions (cycles of length two) in any representation of π as a product of transpositions. One 1Corresponding author. n c way of determining the parity of π is by obtaining the sign of ( 1) − , where c is the − number of cycles in the cycle representation of π. -
Arxiv:2009.00554V2 [Math.CO] 10 Sep 2020 Graph On√ More Than Half of the Vertices of the D-Dimensional Hypercube Qd Has Maximum Degree at Least D
ON SENSITIVITY IN BIPARTITE CAYLEY GRAPHS IGNACIO GARCÍA-MARCO Facultad de Ciencias, Universidad de La Laguna, La Laguna, Spain KOLJA KNAUER∗ Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Spain ABSTRACT. Huang proved that every set of more than halfp the vertices of the d-dimensional hyper- cube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain in- duced subgraphs of maximum degree 1 on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants and contains a sporadic counterexample encoun- tered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are d-regular containing an d induced matching on a 2d−1 -fraction of the vertices. This is largest possible and answers a question of Lehner and Verret. Second, we consider Huang’s lower bound for graphs with subcubes and show that the corre- sponding lower bound is tight for products of Coxeter groups of type An, I2(2k + 1), and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang’s question. -
Homogeneous Systems (1.5) Linear Independence and Dependence (1.7)
Math 20F, 2015SS1 / TA: Jor-el Briones / Sec: A01 / Handout Page 1 of3 Homogeneous systems (1.5) Homogenous systems are linear systems in the form Ax = 0, where 0 is the 0 vector. Given a system Ax = b, suppose x = α + t1α1 + t2α2 + ::: + tkαk is a solution (in parametric form) to the system, for any values of t1; t2; :::; tk. Then α is a solution to the system Ax = b (seen by seeting t1 = ::: = tk = 0), and α1; α2; :::; αk are solutions to the homoegeneous system Ax = 0. Terms you should know The zero solution (trivial solution): The zero solution is the 0 vector (a vector with all entries being 0), which is always a solution to the homogeneous system Particular solution: Given a system Ax = b, suppose x = α + t1α1 + t2α2 + ::: + tkαk is a solution (in parametric form) to the system, α is the particular solution to the system. The other terms constitute the solution to the associated homogeneous system Ax = 0. Properties of a Homegeneous System 1. A homogeneous system is ALWAYS consistent, since the zero solution, aka the trivial solution, is always a solution to that system. 2. A homogeneous system with at least one free variable has infinitely many solutions. 3. A homogeneous system with more unknowns than equations has infinitely many solu- tions 4. If α1; α2; :::; αk are solutions to a homogeneous system, then ANY linear combination of α1; α2; :::; αk is also a solution to the homogeneous system Important theorems to know Theorem. (Chapter 1, Theorem 6) Given a consistent system Ax = b, suppose α is a solution to the system. -
Linear Algebra Review
Linear Algebra Review Kaiyu Zheng October 2017 Linear algebra is fundamental for many areas in computer science. This document aims at providing a reference (mostly for myself) when I need to remember some concepts or examples. Instead of a collection of facts as the Matrix Cookbook, this document is more gentle like a tutorial. Most of the content come from my notes while taking the undergraduate linear algebra course (Math 308) at the University of Washington. Contents on more advanced topics are collected from reading different sources on the Internet. Contents 3.8 Exponential and 7 Special Matrices 19 Logarithm...... 11 7.1 Block Matrix.... 19 1 Linear System of Equa- 3.9 Conversion Be- 7.2 Orthogonal..... 20 tions2 tween Matrix Nota- 7.3 Diagonal....... 20 tion and Summation 12 7.4 Diagonalizable... 20 2 Vectors3 7.5 Symmetric...... 21 2.1 Linear independence5 4 Vector Spaces 13 7.6 Positive-Definite.. 21 2.2 Linear dependence.5 4.1 Determinant..... 13 7.7 Singular Value De- 2.3 Linear transforma- 4.2 Kernel........ 15 composition..... 22 tion.........5 4.3 Basis......... 15 7.8 Similar........ 22 7.9 Jordan Normal Form 23 4.4 Change of Basis... 16 3 Matrix Algebra6 7.10 Hermitian...... 23 4.5 Dimension, Row & 7.11 Discrete Fourier 3.1 Addition.......6 Column Space, and Transform...... 24 3.2 Scalar Multiplication6 Rank......... 17 3.3 Matrix Multiplication6 8 Matrix Calculus 24 3.4 Transpose......8 5 Eigen 17 8.1 Differentiation... 24 3.4.1 Conjugate 5.1 Multiplicity of 8.2 Jacobian...... -
Notes on Finite Group Theory
Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geome- try (where groups describe in a very detailed way the symmetries of geometric objects) and in the theory of polynomial equations (developed by Galois, who showed how to associate a finite group with any polynomial equation in such a way that the structure of the group encodes information about the process of solv- ing the equation). These notes are based on a Masters course I gave at Queen Mary, University of London. Of the two lecturers who preceded me, one had concentrated on finite soluble groups, the other on finite simple groups; I have tried to steer a middle course, while keeping finite groups as the focus. The notes do not in any sense form a textbook, even on finite group theory. Finite group theory has been enormously changed in the last few decades by the immense Classification of Finite Simple Groups. The most important structure theorem for finite groups is the Jordan–Holder¨ Theorem, which shows that any finite group is built up from finite simple groups. If the finite simple groups are the building blocks of finite group theory, then extension theory is the mortar that holds them together, so I have covered both of these topics in some detail: examples of simple groups are given (alternating groups and projective special linear groups), and extension theory (via factor sets) is developed for extensions of abelian groups. In a Masters course, it is not possible to assume that all the students have reached any given level of proficiency at group theory. -
A SHORT PROOF of the EXISTENCE of JORDAN NORMAL FORM Let V Be a Finite-Dimensional Complex Vector Space and Let T : V → V Be A
A SHORT PROOF OF THE EXISTENCE OF JORDAN NORMAL FORM MARK WILDON Let V be a finite-dimensional complex vector space and let T : V → V be a linear map. A fundamental theorem in linear algebra asserts that there is a basis of V in which T is represented by a matrix in Jordan normal form J1 0 ... 0 0 J2 ... 0 . . .. . 0 0 ...Jk where each Ji is a matrix of the form λ 1 ... 0 0 0 λ . 0 0 . . .. 0 0 . λ 1 0 0 ... 0 λ for some λ ∈ C. We shall assume that the usual reduction to the case where some power of T is the zero map has been made. (See [1, §58] for a characteristically clear account of this step.) After this reduction, it is sufficient to prove the following theorem. Theorem 1. If T : V → V is a linear transformation of a finite-dimensional vector space such that T m = 0 for some m ≥ 1, then there is a basis of V of the form a1−1 ak−1 u1, T u1,...,T u1, . , uk, T uk,...,T uk a where T i ui = 0 for 1 ≤ i ≤ k. At this point all the proofs the author has seen (even Halmos’ in [1, §57]) become unnecessarily long-winded. In this note we present a simple proof which leads to a straightforward algorithm for finding the required basis. Date: December 2007. 1 2 MARK WILDON Proof. We work by induction on dim V . For the inductive step we may assume that dim V ≥ 1. -
Lecture Notes to Be Used in Conjunction with 233 Computational
Lecture Notes to be used in conjunction with 233 Computational Techniques Istv´an Maros Department of Computing Imperial College London V2.9e January 2008 CONTENTS i Contents 1 Introduction 1 2 Computation and Accuracy 1 2.1 MeasuringErrors ..................... 1 2.2 Floating-Point RepresentationofNumbers . 2 2.3 ArithmeticOperations . 5 3 Computational Linear Algebra 6 3.1 VectorsandMatrices . 6 3.1.1 Vectors ....................... 6 3.1.2 Matrices ...................... 10 3.1.3 More about vectors and matrices ........... 14 3.2 Norms .......................... 20 3.2.1 Vector norms .................... 21 3.2.2 Matrix norms .................... 23 3.3 VectorSpaces....................... 25 3.3.1 Linear dependence and independence ......... 25 3.3.2 Range and null spaces ................ 30 3.3.3 Singular and nonsingular matrices ........... 35 3.4 Eigenvalues, Eigenvectors, Singular Values . 37 3.4.1 Eigenvalues and eigenvectors ............. 37 3.4.2 Spectral decomposition ............... 39 3.4.3 Singular value decomposition ............. 41 3.5 Solution of Systems of Linear Equations . 41 3.5.1 Triangular systems .................. 42 3.5.2 Gaussian elimination ................. 44 3.5.3 Symmetric systems, Cholesky factorization ....... 52 CONTENTS ii 3.5.4 Gauss-Jordan elimination if m<n ........ 56 3.6 Solution of Systems of Linear Differential Equations . 59 3.7 LeastSquaresProblems. 65 3.7.1 Formulation of the least squares problem ........ 65 3.7.2 Solution of the least squares problem ......... 67 3.8 ConditionofaMathematicalProblem . 71 3.8.1 Matrix norms revisited ................ 72 3.8.2 Condition of a linear system with a nonsingular matrix . 73 3.8.3 Condition of the normal equations ........... 78 3.9 Introduction to Sparse Computing .