DOCSLIB.ORG

Things You Need to Know About Linear Algebra Math 131 Multivariate

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. valued functions. So, here, a and b are vectors, that is, points, in Rn, while c is a scalar, that is, a Section 1.3. The length kak of a vector a (also real number in R, and f is a vector-valued function. called its norm). The triangle inequality. In multivariate calculus, we'll look at nonlinear functions f : Rn ! Rm from n-dimensional space ka − bk ≤ kak + kbk to m-dimensional space. We'll restrict ourselves, however, to those that can be differentiated, or at b least continuous, just about everywhere. ¡HH ¡ H In order to study these nonlinear functions, we'll HH ¡ HH use the linear functions from linear algebra. Thus, ¡ Hj 1 a you'll need to know quite a bit about linear algebra. ¡ ¡ ¡ Topics for review. Here's a list of the topics ¡ 0 we'll review. The text covers them all in detail, and we'll look at nearly all of them, but concentrate on Dot products. Dot products a · b of vectors (some- the most important ones. times called inner products and denoted hajib). We'll use most of these concepts as early as chap- Law of cosines and angles. Dot product of two vec- ter 2, the chapter on derivatives of multivariate tors in terms of the angle between them functions. A few we won't use until chapter 3, but we'll need all of them by the end of the course. a · b = kak kbk cos θ: Section 1.1. Vectors in two and three dimen- Two vectors are orthogonal, a ? b, if and only if sions. The standard (x; y) coordinate plane, and their dot product is 0. The projection projab of the 1 vector b onto the vector a is point and a line. Distance between parallel planes. Distance between skew lines. a · b Occasionally, we'll use these geometric concepts, projab = 2 a: kak especially in examples. Unit vectors. Normalization of vectors. The concept of length of a vector is very impor- Section 1.6. Summary of geometry in n-space tant, especially in the form kb − ak which gives the Rn. The Cauchy inequality and the triangle in- distance between two points b and a. In chapter equality. The standard basis fe1; e2;:::; eng for 2 we'll define the limit of a function f : Rn ! Rm Rn. m × n matrices as descriptions of linear func- using this concept. In particular, we'll define tions f : Rn ! Rm. Composition of these functions justifies matrix multiplication. Row and column lim f(x) = L vectors. Properties of matrix operations. Hyper- x!a planes. Determinants, minors, and cofactors. A as saying that we can make kf(x) − Lk arbitrarily square matrix is invertible when its determinant is small (less than ) by keeping kx − ak sufficiently nonzero. small (less than δ). In chapter 2 we'll define the derivative of a func- Later in chapter 2 we'll see how directional tion f : Rn ! Rm as the m × n matrix of partial derivatives can be found in terms of dot products, derivatives. Later in that chapter, we'll see that the and we'll find tangent planes (and hyperplanes) in chain rule describes the derivative of a composition terms of dot products. We'll frequently use length of two functions as the product of the two matrices and dot products throughout the course. describing the derivatives of the two functions. In one section of chapter 3 we'll study arc length Also in chapter 2 we'll see how determinants are and curvature, and in that section we'll see how unit used in the inverse and implicit function theorems. vectors and normalization can be used to simplify We'll use matrices, vectors, and matrix products their analysis. beginning in chapter 3 when we study the Hessian of a function. Determinants are also used in chapter Section 1.4. Cross products. Cross products a× 3 when we look at curl. b of pairs of vectors in R3. Areas of parallelograms and triangles in terms of cross products. Matrices Section 1.7. Coordinate systems. Rectangular and determinants as related to cross products. The coordinates. Polar coordinates for planes. Conver- triple scalar product, volume of a parallelepiped. sion between rectangular and polar coordinates for Rotation and angular velocity. the plane. Cylindrical and spherical coordinates for Cross products and dot products together allow space. Conversion among the various coordinate us to study geometry in three dimensions. In chap- systems for space. ter 3 we'll look at Kepler's laws of planetary motion We'll use these other coordinate systems in many using derivatives of cross products and dot products of the examples we examine throughout the course. of functions. Later in the chapter, we'll study the gradient, divergence, and curl of scalar and vector Math 131 Home Page at fields. These concepts depend on cross products http://math.clarku.edu/~djoyce/ma131/ and dot products. Section 1.5. Equations and distance in R3. Equations of planes in three space including para- metric equations of planes. Distance between a 2.
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • MATH 532: Linear Algebra Chapter 4: Vector Spaces
    MATH 532: Linear Algebra Chapter 4: Vector Spaces Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2015 [email protected] MATH 532 1 Outline 1 Spaces and Subspaces 2 Four Fundamental Subspaces 3 Linear Independence 4 Bases and Dimension 5 More About Rank 6 Classical Least Squares 7 Kriging as best linear unbiased predictor [email protected] MATH 532 2 Spaces and Subspaces Outline 1 Spaces and Subspaces 2 Four Fundamental Subspaces 3 Linear Independence 4 Bases and Dimension 5 More About Rank 6 Classical Least Squares 7 Kriging as best linear unbiased predictor [email protected] MATH 532 3 Spaces and Subspaces Spaces and Subspaces While the discussion of vector spaces can be rather dry and abstract, they are an essential tool for describing the world we work in, and to understand many practically relevant consequences. After all, linear algebra is pretty much the workhorse of modern applied mathematics. Moreover, many concepts we discuss now for traditional “vectors” apply also to vector spaces of functions, which form the foundation of functional analysis. [email protected] MATH 532 4 Spaces and Subspaces Vector Space Definition A set V of elements (vectors) is called a vector space (or linear space) over the scalar field F if (A1) x + y 2 V for any x; y 2 V (M1) αx 2 V for every α 2 F and (closed under addition), x 2 V (closed under scalar (A2) (x + y) + z = x + (y + z) for all multiplication), x; y; z 2 V, (M2) (αβ)x = α(βx) for all αβ 2 F, (A3) x + y = y + x for all x; y 2 V, x 2 V, (A4) There exists a zero vector 0 2 V (M3) α(x + y) = αx + αy for all α 2 F, such that x + 0 = x for every x; y 2 V, x 2 V, (M4) (α + β)x = αx + βx for all (A5) For every x 2 V there is a α; β 2 F, x 2 V, negative (−x) 2 V such that (M5)1 x = x for all x 2 V.
    [Show full text]
  • On the History of Some Linear Algebra Concepts:From Babylon to Pre
    The Online Journal of Science and Technology - January 2017 Volume 7, Issue 1 ON THE HISTORY OF SOME LINEAR ALGEBRA CONCEPTS: FROM BABYLON TO PRE-TECHNOLOGY Sinan AYDIN Kocaeli University, Kocaeli Vocational High school, Kocaeli – Turkey [email protected] Abstract: Linear algebra is a basic abstract mathematical course taught at universities with calculus. It first emerged from the study of determinants in 1693. As a textbook, linear algebra was first used in graduate level curricula in 1940’s at American universities. In the 2000s, science departments of universities all over the world give this lecture in their undergraduate programs. The study of systems of linear equations first introduced by the Babylonians around at 1800 BC. For solving of linear equations systems, Cardan constructed a simple rule for two linear equations with two unknowns around at 1550 AD. Lagrange used matrices in his work on the optimization problems of real functions around at 1750 AD. Also, determinant concept was used by Gauss at that times. Between 1800 and 1900, there was a rapid and important developments in the context of linear algebra. Some theorems for determinant, the concept of eigenvalues, diagonalisation of a matrix and similar matrix concept were added in linear algebra by Couchy. Vector concept, one of the main part of liner algebra, first used by Grassmann as vector product and vector operations. The term ‘matrix’ as a rectangular forms of scalars was first introduced by J. Sylvester. The configuration of linear transformations and its connection with the matrix addition, multiplication and scalar multiplication were studied first by A.
    [Show full text]
  • Multi-Variable Calculus/Linear Algebra Scope & Sequence
    Multi-Variable Calculus/Linear Algebra Scope & Sequence Grading Period Unit Title Learning Targets Throughout the *Apply mathematics to problems in everyday life School Year *Use a problem-solving model that incorporates analyzing information, formulating a plan, determining a solution, justifying the solution and evaluating the reasonableness of the solution *Select tools to solve problems *Communicate mathematical ideas, reasoning and their implications using multiple representations *Create and use representations to organize, record and communicate mathematical ideas *Analyze mathematical relationships to connect and communicate mathematical ideas *Display, explain and justify mathematical ideas and arguments First Grading Vectors in a plane and Unit vectors, graphing and basic operations Period in space MULTIVARIABLE Dot products and cross Angles between vectors, orthogonality, projections, work CALCULUS products Lines and Planes in Equations, intersections, distance space Surfaces in Space Cylindrical surfaces, quadric surfaces, surfaces of revolution, cylindrical and spherical coordinate Vector Valued Parameterization, domain, range, limits, graphical representation, continuity, differentiation, Functions integration, smoothness, velocity, acceleration, speed, position for a projectile, tangent and normal vectors, principal unit normal, arc length and curvature. Functions of several Graphically, level curves, delta neighborhoods, interior point, boundary point, closed regions, variables limits by definition, limits from various
    [Show full text]
  • MATH 304 Linear Algebra Lecture 6: Transpose of a Matrix. Determinants. Transpose of a Matrix
    MATH 304 Linear Algebra Lecture 6: Transpose of a matrix. Determinants. Transpose of a matrix Definition. Given a matrix A, the transpose of A, denoted AT , is the matrix whose rows are columns of A (and whose columns are rows of A). That is, T if A = (aij ) then A = (bij ), where bij = aji . T 1 4 1 2 3 Examples. = 2 5, 4 5 6 3 6 T 7 T 4 7 4 7 8 = (7, 8, 9), = . 7 0 7 0 9 Properties of transposes: • (AT )T = A • (A + B)T = AT + BT • (rA)T = rAT • (AB)T = BT AT T T T T • (A1A2 ... Ak ) = Ak ... A2 A1 • (A−1)T = (AT )−1 Definition. A square matrix A is said to be symmetric if AT = A. For example, any diagonal matrix is symmetric. Proposition For any square matrix A the matrices B = AAT and C = A + AT are symmetric. Proof: BT = (AAT )T = (AT )T AT = AAT = B, C T = (A + AT )T = AT + (AT )T = AT + A = C. Determinants Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (aij )1≤i,j≤n is denoted det A or a11 a12 ... a1n a a ... a 21 22 2n . .. an1 an2 ... ann Principal property: det A =0 if and only if the matrix A is singular. Definition in low dimensions a b Definition. det (a) = a, = ad − bc, c d a a a 11 12 13 a21 a22 a23 = a11a22a33 + a12a23a31 + a13a21a32− a31 a32 a33 −a13a22a31 − a12a21a33 − a11a23a32. * ∗ ∗ ∗ * ∗ ∗ ∗ * +: ∗ * ∗ , ∗ ∗ * , * ∗ ∗ . ∗ ∗ * * ∗ ∗ ∗ * ∗ ∗ ∗ * ∗ * ∗ * ∗ ∗ − : ∗ * ∗ , * ∗ ∗ , ∗ ∗ * . * ∗ ∗ ∗ ∗ * ∗ * ∗ Examples: 2×2 matrices 1 0 3 0 = 1, = − 12, 0 1 0 −4 −2 5 7 0 = − 6, = 14, 0 3 5 2 0 −1 0 0 = 1, = 0, 1 0 4 1 −1 3 2 1 = 0, = 0.
    [Show full text]
  • 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 0 0 .. λ 1 0 0 0 ··· 0 λ A square matrix B is in the Jordan canonical form if it has diagonal block structure J1 O . O O J2 . O B = . , . .. O O . Jk where each diagonal block Ji is a Jordan block. Consider an n×n Jordan block λ 1 0 ··· 0 0 . 0 λ 1 .. 0 0 . 0 0 λ .. 0 0 J = . . .. .. . 0 0 0 .. λ 1 0 0 0 ··· 0 λ Multiplication by J − λI acts on the standard basis by a chain rule: 0 e1 e2 en−1 en ◦ ←− • ←− •←−··· ←− • ←− • 2 1 0000 0 2 0000 0 0 3 0 0 0 Consider a matrix B = . 0 0 0 3 1 0 0 0 0 0 3 1 0 0 0 0 0 3 This matrix is in Jordan canonical form. Vectors from the standard basis are organized in several chains. Multiplication by B − 2I : 0 e1 e2 ◦ ←− • ←− • Multiplication by B − 3I : 0 e3 ◦ ←− • 0 e4 e5 e6 ◦ ←− • ←− • ←− • Generalized eigenvectors Let L : V → V be a linear operator on a vector space V . Definition. A nonzero vector v is called a generalized eigenvector of L associated with an eigenvalue λ if (L − λI)k (v)= 0 for some integer k ≥ 1 (here I denotes the identity map on V ). The set of all generalized eigenvectors for a particular λ along with the zero vector is called the generalized eigenspace and denoted Kλ.
    [Show full text]
  • Eigenvalues, Eigenvectors, and Eigenspaces of Linear Operators
    Therefore x + cy is also a λ-eigenvector. Thus, the set of λ-eigenvectors form a subspace of F n. q.e.d. One reason these eigenvalues and eigenspaces are important is that you can determine many of the Eigenvalues, eigenvectors, and properties of the transformation from them, and eigenspaces of linear operators that those properties are the most important prop- Math 130 Linear Algebra erties of the transformation. D Joyce, Fall 2015 These are matrix invariants. Note that the Eigenvalues and eigenvectors. We're looking eigenvalues, eigenvectors, and eigenspaces of a lin- at linear operators on a vector space V , that is, ear transformation were defined in terms of the linear transformations x 7! T (x) from the vector transformation, not in terms of a matrix that de- space V to itself. scribes the transformation relative to a particu- When V has finite dimension n with a specified lar basis. That means that they are invariants of basis β, then T is described by a square n×n matrix square matrices under change of basis. Recall that A = [T ]β. if A and B represent the transformation with re- We're particularly interested in the study the ge- spect to two different bases, then A and B are con- ometry of these transformations in a way that we jugate matrices, that is, B = P −1AP where P is can't when the transformation goes from one vec- the transition matrix between the two bases. The tor space to a different vector space, namely, we'll eigenvalues are numbers, and they'll be the same compare the original vector x to its image T (x).
    [Show full text]
  • Generalized Eigenvector - Wikipedia
    11/24/2018 Generalized eigenvector - Wikipedia Generalized eigenvector In linear algebra, a generalized eigenvector of an n × n matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.[1] Let be an n-dimensional vector space; let be a linear map in L(V), the set of all linear maps from into itself; and let be the matrix representation of with respect to some ordered basis. There may not always exist a full set of n linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.[2][3] This happens when the algebraic multiplicity of at least one eigenvalue is greater than its geometric multiplicity (the nullity of the matrix , or the dimension of its nullspace). In this case, is called a defective eigenvalue and is called a defective matrix.[4] A generalized eigenvector corresponding to , together with the matrix generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of .[5][6][7] Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for .[8] This basis can be used to determine an "almost diagonal matrix" in Jordan normal form, similar to , which is useful in computing certain matrix functions of .[9] The matrix is also useful in solving the system of linear differential equations where need not be diagonalizable.[10][11] Contents Overview and definition Examples Example 1 Example 2 Jordan chains
    [Show full text]
  • Generalized Eigenvectors, Minimal Polynomials and Theorem of Cayley-Hamiltion
    GENERALIZED EIGENVECTORS, MINIMAL POLYNOMIALS AND THEOREM OF CAYLEY-HAMILTION FRANZ LUEF Abstract. Our exposition is inspired by S. Axler’s approach to linear algebra and follows largely his exposition in ”Down with Determinants”, check also the book ”LinearAlgebraDoneRight” by S. Axler [1]. These are the lecture notes for the course of Prof. H.G. Feichtinger ”Lineare Algebra 2” from 15.11.2006. Before we introduce generalized eigenvectors of a linear transformation we recall some basic facts about eigenvalues and eigenvectors of a linear transformation. Let V be a n-dimensional complex vector space. Recall a complex number λ is called an eigenvalue of a linear operator T on V if T − λI is not injective, i.e. ker(T − λI) 6= {0}. The main result about eigenvalues is that every linear op- erator on a finite-dimensional complex vector space has an eigenvalue! Furthermore we call a vector v ∈ V an eigenvector of T if T v = λv for some eigenvalue λ. The central result on eigenvectors is that Non-zero eigenvectors corresponding to distinct eigenvalues of a linear transformation on V are linearly independent. Consequently the number of distinct eigenvalues of T can- not exceed thte dimension of V . Unfortunately the eigenvectors of T need not span V . For example the linear transformation on C4 whose matrix is 0 1 0 0 0 0 1 0 T = 0 0 0 1 0 0 0 0 as only the eigenvalue 0, and its eigenvectors form a one-dimensional subspace of C4. Observe that T,T 2 6= 0 but T 3 = 0.
    [Show full text]
  • Melissa Mogensen History of Mathematics History of Linear
    Melissa Mogensen History of Mathematics History of Linear Algebra Timeline 1900 BC- Babylon The beginnings of matrices and determinants arise through the study of systems of linear equations, which was first introduced by the Babylonians. The Babylonians studied problems that led to simultaneous linear equations. 200 BC- China The Chinese actually came a lot closer to actual matrices we use today by using a method that resembles Gaussian Elimination. This technique was used before they had equations or variables. 1545 AD- Cardan Cardon, in Ars Magna, gives a rule for solving a system of two linear equations called regula de modo, which is essentially Cramer’s rule, but he (Cardan) doesn’t make the final step, so he doesn’t reach the definition of a determinant. 1683 AD- Seki In Japan, Seki wrote Method of solving the dissimulated problems, which contains matrix methods written in tablets, the exact way the Chinese recorded them. Seki also introduced determinants and gave general methods for calculating them based on examples. He was able to find up to 5X5 matrices and applied them to solving equations but not systems of equations. 1683 AD- Leibnitz In Europe, Leibnitz explained a system of equations 10+11x+12y=0 20+21x+22y=0 had a solution because 30+31x+32y=0 10.21.32 + 11.22.30 + 12.20.31 = 10.22.31 + 11.20.32 + 12.21.30 which is exactly the condition that the coefficient matrix has determinant 0. The periods resemble multiplication and the two characters describe first the equation and the second resembles the letter.
    [Show full text]