Chapter 4: Binary Operations and Relations
|
|
|
- Esmond James
- 7 years ago
- Views:
Transcription
1 c Dr Oksana Shatalov, Fall Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction, multiplication are binary operations on Z. Addition is a binary operation on Q because Division is NOT a binary operation on Z because Division is a binary operation on Classification of binary operations by their properties Associative and Commutative Laws DEFINITION 2. A binary operation on A is associative if a, b, c A, (a b) c = a (b c). A binary operation on A is commutative if Identities a, b A, a b = b a. DEFINITION 3. If is a binary operation on A, an element e A is an identity element of A w.r.t if a A, a e = e a = a. EXAMPLE 4. 1 is an identity element for Z, Q and R w.r.t. multiplication. 0 is an identity element for Z, Q and R w.r.t. addition.
2 c Dr Oksana Shatalov, Fall Inverses DEFINITION 5. Let be a binary operation on A with identity e, and let a A. We say that a is invertible w.r.t. if there exists b A such that a b = b a = e. If f exists, we say that b is an inverse of a w.r.t. and write b = a 1. Note, inverses may or may not exist. EXAMPLE 6. Every x Z has inverse w.r.t. addition because x Z, x + ( x) = ( x) + x = 0. However, very few elements in Z have multiplicative inverses. Namely, EXAMPLE 7. Let be a binary operation on Z defined by (a) Prove that the operation is binary. a, b Z, a b = a + 3b 1. (b) Determine whether the operation is associative and/or commutative. Prove your answers. (c) Determine whether the operation has identities. (d) Discuss inverses.
3 c Dr Oksana Shatalov, Fall EXAMPLE 8. Let be a binary operation on the power set P (A) defined by (a) Prove that the operation is binary. X, Y P (A), X Y = X Y. (b) Determine whether the operation is associative and/or commutative. Prove your answers. (c) Determine whether the operation has identities. (d) Discuss inverses. EXAMPLE 9. Let be a binary operation on F (A) defined by (a) Prove that the operation is binary. f, g F (A), f g = f g. (b) Determine whether the operation is associative and/or commutative. Prove your answers.
4 c Dr Oksana Shatalov, Fall (c) Determine whether the operation has identities. (d) Discuss inverses. EXAMPLE Let be a binary operation on the set M 2 (R) of all 2 2 matrices defined by (a) Prove that the operation is binary. A 1, A 2 M 2 (R), A 1 A 2 = A 1 + A 2. (b) Determine whether the operation is associative and/or commutative. Prove your answers. 1 se Appendix at the end of the Chapter.
5 c Dr Oksana Shatalov, Fall (c) Determine whether the operation has identities. (d) Discuss inverses. EXAMPLE 11. Let be a binary operation on the set M 2 (R) of all 2 2 matrices defined by (a) Prove that the operation is binary. A 1, A 2 M 2 (R), A 1 A 2 = A 1 A 2. (b) Determine whether the operation is associative and/or commutative. Prove your answers.
6 c Dr Oksana Shatalov, Fall (c) Show that the matrix I = ( ) is an identity element w.r.t.. (d) Discuss inverses (Use the following FACT: A matrix is invertible if and only if its derminant does not equal to zero ). PROPOSITION 12. Let be a binary operation on a nonempty set A. If e is an identity element on A then e is unique. Proof. PROPOSITION 13. Let be an associative binary operation on a nonempty set A with the identity e, and if a A has an inverse element w.r.t., then this inverse element is unique. Proof. See Exercise 12.
7 c Dr Oksana Shatalov, Fall Closure DEFINITION 14. Let be a binary operation on a nonempty set A, and suppose that X A. If is also a binary operation on X then we say that X is closed in A under. EXAMPLE 15. Determine whether the following subsets of Z are closed in Z under addition and multiplication. (a) Z + (b) E (c) O EXAMPLE 16. Determine whether the following subsets of M 2 (R) is closed in M 2 (R) under matrix addition and multiplication: {( ) } a b S = M c d 2 (R) a = d.
8 c Dr Oksana Shatalov, Fall : Equivalence Relations DEFINITION 17. A relation R on a set A is a subset of A A. If (a, b) R, we write arb. EXAMPLE 18. On the set R one can define arb by a < b. Then, for example, EXAMPLE 19. On the power set P (Z) one can define R by ARB if A = B. Properties of Relations DEFINITION 20. Let R be a relation on a set A. We say: 1. R is reflexive if ara, a A. 2. R is symmetric if a, b A, if arb then bra. 3. R is transitive if a, b, c A, if arb and brc, then arc. 4. R is antisymmetric if a, b A, if arb and bra, ten a = b.. DEFINITION 21. A relation R on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. EXAMPLE 22. Let R be the relation on Z defined by arb if a b. Determine whether it is reflexive, symmetric, transitive, or antisymmetric.
9 c Dr Oksana Shatalov, Fall EXAMPLE 23. Let R be the relation on R defined by arb if a b 1 (that is a is related to b if the distance between a and b is at most 1.) Determine whether it is reflexive, symmetric, transitive, or antisymmetric. EXAMPLE 24. Let R be the relation on Z defined by arb if a + 3b E. Show that R is an equivalence relation. REMARK 25. When R is an equivalence relation, it is common to write a b instead of arb, read a is equivalent to b.
10 c Dr Oksana Shatalov, Fall EXAMPLE 26. Let n Z +. Define arb on Z by n a b. (In particular, if n = 2 the arb means a b is ). Show that R is an equivalence relation. REMARK 27. The above relation is called congruence mod n, and usually written a b (mod n)
11 c Dr Oksana Shatalov, Fall Equivalence Classes DEFINITION 28. If R is an equivalence relation on a set A, and a A, then the set [a] = {x A x a} is called the equivalence class of a. Elements of the same class are said to be equivalent. EXAMPLE 29. Define arb on Z by 2 a b. (In other words, R is the relation of congruence mod 2 on Z.) (a) What integers are in the equivalence class of 6? (b) What integers are in the equivalence class of 25? (c) How many distinct equivalence classes there? What are they? EXAMPLE 30. Define arb on Z by n a b. (In other words, R is the relation of congruence mod n on Z.) (a) How many distinct equivalence classes there? What are they?
12 c Dr Oksana Shatalov, Fall (b) Show that the set of these equivalence classes forms a partition of Z. THEOREM 31. If R is an equivalence relation on a nonempty set A, then the set of equivalence classes on R forms a partition on A. Proof. So, any equivalence relation on a set A leads to a partition of A. In addition, any partition of A gives rise to an equivalence relation on A.
13 c Dr Oksana Shatalov, Fall THEOREM 32. Let R be a partition of a nonempty set A. Define a relation R 1 on A by ar 1 b if a and b are in the same element of the partition R. Then R 1 is an equivalence relation on A. Proof. Conclusion: Theorems 31 and 32 imply that there is a bijection between the set of all equivalence relations of A and the set of all partitions on A. EXAMPLE 33. Let R be the relation on Z defined by arb if a + 3b E. By one of the above examples, R is an equivalence relation. Determine all equivalence classes for R.
14 c Dr Oksana Shatalov, Fall Partial and linear ordering Recall that arb defined by a b, a, b R, is not an equivalence relation. Why? DEFINITION 34. A relation R on a set A is called a partial ordering on A if R is reflexive, transitive and antisymmetric. If A is a set and there exists a partial ordering on A, then we say that A is a partially ordered set. EXAMPLE 35. For all X, Y P (A) define R by X Y. Then R is a partial ordering of P (A). DEFINITION 36. Let A be a set and R be a partial ordering on A. We say that R is a linear ordering on A if for all a, b A either arb, or bra. EXAMPLE 37. is a linear ordering of R EXAMPLE 38. Discuss when the relation from Example 35 is a linear ordering.
15 c Dr Oksana Shatalov, Fall EXAMPLE 39. (cf. Example 17(c).) Let R be a relation on a set A. If R is both symmetric and antisymmetric, does it follow that R is reflexive?
16 c Dr Oksana Shatalov, Fall Appendix: Matrices and Matrix Multiplication 1. An m n (this is often called the size or dimension of the matrix) matrix is a table with m rows and n columns and the entry in the i-th row and j-th column is denoted by a ij : a 11 a a 1n a 21 a a 2n.... a m1 a m2... a mn 2. A matrix is usually denoted by a capital letter and its elements by small letters : a ij = entry in the ith row and jth column of A. 3. Two matrices are said to be equal if they are the same size and each corresponding entry is equal. 4. Special Matrices: A square matrix is any matrix whose size (or dimension) is n n (i.e. it has the same number of rows as columns.) In a square matrix the diagonal that starts in the upper left and ends in the lower right is often called the main diagonal. The zero matrix is a matrix all of whose entries are zeroes. The identity matrix is a square n n matrix, denoted I n, whose main diagonal consists of all 1 s and all the other elements are zero: I n = The diagonal matrix is a square n n matrix of the following form λ λ λ n Column matrix (=column vector) and the row matrix (=row vector) are those matrices that consist of a single column or a single row respectively: x 1 x 2 X =., Y = ( y 1 y 2... ) y n x n Note that an n-dimensional column vector is an n 1 matrix, and an n-dimensional row vector is an 1 n matrix. Transpose of a Matrix: If A is an m n matrix with entries a ij, then A T is the n m matrix with entries a ji. A T is obtained by interchanging rows and columns of A.
17 c Dr Oksana Shatalov, Fall Matrix Arithmetic The sum or difference of two matrices of the same size is a new matrix of identical size whose entries are the sum or difference of the corresponding entries from the original two matrices. Note that we cant add or subtract entries with different sizes. The scalar multiplication multiplied by that constant. If A, B, and C are matrices of the same size, then (a) A + B = B + A (Commutative Law) (b) (A + B) + C = A + (B + C) (Associative Law) by a constant gives a new matrix whose entries have all been Matrix multiplication: If Y is a row matrix of size 1 n and X is a column matrix of size n 1 (see above), then the matrix product of Y and X is defined by x 1 Y X = ( y 1 y 2 y 3 ) x 2 y n x 3 = y 1 x 1 + y 2 x 2 + y 3 x y n x n. x n If A is an m p matrix and matrix B is p n, then the product AB is an m n matrix, and its element in the ith row and jth column is the product of the ith row of A and the jth column of B. RULE for multiplying matrices: The column of the 1st matrix must be the same size as the row of the 2nd matrix. 6. Example. (a) Given Compute A 2B A = 3 4, B = (b) Let A = ( ) and B = ( ). Find BA T.
18 c Dr Oksana Shatalov, Fall (c) Example. Given 1 2 A = 3 4, B = 1 2 Compute AB and BA when it is possible. ( ) (d) Compute FACTS and LAWS FOR MATRIX MULTIPLICATION: If the size requirements are met for matrices A, B and C, then AB BA (NOT always Commutative) 2 A(B + C) = AB + AC (always Distributive) (AB)C = A(BC) (always Associative) AB = 0 does not imply that A = 0 or B = 0. AB = AC does not imply that B = C. I n A = AI n = A for any square matrix A of size n. Determinant 8. Determinant of a matrix is a function that takes a square matrix and converts it into a number. 9. Determinant of 2 2 matrix: a c b d = ad bc 10. Matrix Inverse. Let A be a square matrix of size n. A square matrix, A 1, of size n, such that AA 1 = I n (or, equivalently, A 1 A = I n ) is called an inverse matrix. ( ) 11. A 1 a11 a in the case n = 2: If A = 12 then a 21 a 22 A 1 = 1 ( ) a22 a 12 deta a 21 a FACT: A 1 exists if and only if det A 0. 2 Since the multiplication of matrices is NOT commutative, you MUST multiply left to right.
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 +
Lecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
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
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 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
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 Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
Math 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
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
Mathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
Using row reduction to calculate the inverse and the determinant of a square matrix
Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible
Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
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
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)(
Notes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
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
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
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
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
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
NOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
GENERATING SETS KEITH CONRAD
GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors
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
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
Solution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
Introduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
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
Lecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
[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:
T ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
Lecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
Matrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
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
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
Inner products on R n, and more
Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +
Vector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
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
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
Matrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
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
Unit 18 Determinants
Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of
Properties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
Elementary Matrices and The LU Factorization
lementary Matrices and The LU Factorization Definition: ny matrix obtained by performing a single elementary row operation (RO) on the identity (unit) matrix is called an elementary matrix. There are three
Direct Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
Solving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
The Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are
160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
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
Subspaces of R n LECTURE 7. 1. Subspaces
LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r
Solving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
Lecture 4: Partitioned Matrices and Determinants
Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying
Linear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
DETERMINANTS TERRY A. LORING
DETERMINANTS TERRY A. LORING 1. Determinants: a Row Operation By-Product The determinant is best understood in terms of row operations, in my opinion. Most books start by defining the determinant via formulas
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,
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
Notes on Linear Algebra. Peter J. Cameron
Notes on Linear Algebra Peter J. Cameron ii Preface Linear algebra has two aspects. Abstractly, it is the study of vector spaces over fields, and their linear maps and bilinear forms. Concretely, it is
Elements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
Systems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
26 Ideals and Quotient Rings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed
Notes on Algebraic Structures. Peter J. Cameron
Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester
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
Solutions to In-Class Problems Week 4, Mon.
Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 26 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 26, 2005, 1050 minutes Solutions
Brief Introduction to Vectors and Matrices
CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vector-valued
Lecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
Matrix Algebra in R A Minimal Introduction
A Minimal Introduction James H. Steiger Department of Psychology and Human Development Vanderbilt University Regression Modeling, 2009 1 Defining a Matrix in R Entering by Columns Entering by Rows Entering
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
Notes on Symmetric Matrices
CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.
Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
So let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
Linear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
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
I. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
DATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
Chapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
LINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
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
( ) 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
A Primer on Index Notation
A Primer on John Crimaldi August 28, 2006 1. Index versus Index notation (a.k.a. Cartesian notation) is a powerful tool for manipulating multidimensional equations. However, there are times when the more
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
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
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
4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
Discrete Mathematics. Hans Cuypers. October 11, 2007
Hans Cuypers October 11, 2007 1 Contents 1. Relations 4 1.1. Binary relations................................ 4 1.2. Equivalence relations............................. 6 1.3. Relations and Directed Graphs.......................
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
6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
Linear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
Boolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
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
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
Applications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
Abstract Algebra Cheat Sheet
Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the