Section Summary. Sequences. Recurrence Relations. Summations Special Integer Sequences (optional)

Size: px
Start display at page:

Download "Section Summary. Sequences. Recurrence Relations. Summations Special Integer Sequences (optional)"

Transcription

1 Section 2.4

2 Section Summary Sequences. o Examples: Geometric Progression, Arithmetic Progression Recurrence Relations o Example: Fibonacci Sequence Summations Special Integer Sequences (optional)

3 Sequences Definition1: A sequence is a function from a subset of the integers (usually either the set {0, 1, 2, 3, 4,..} or {1, 2, 3, 4,.} ) to a set S. The notation a n is used to denote the image of the integer n. We can think of a n as the equivalent of f(n) where f is a function from {0,1,2,..} to S. We call a n a term of the sequence. o A sequence is a discrete structure used to represent an ordered list. o 1,2,3,5,8 is a finite sequence, 1,3,9,27,,3 n, is an finite sequence Example: consider the sequence {a n }, where a n =1/n, list the first four items of the sequence. Solution: o 1,1/2,1/3,1/4

4 Geometric Progression Definition 2: A geometric progression is a sequence of the form: where the initial term a and the common ratio r are real numbers. Examples: 1. Let a = 1 and r = 1. Then: 2. Let a = 2 and r = 5. Then: 3. Let a = 6 and r = 1/3. Then:

5 Arithmetic Progression Definition 3: A arithmetic progression is a sequence of the form: where the initial term a and the common difference d are real numbers. Examples: 1. Let a = 1 and d = 4: 2. Let a = 7 and d = 3: 3. Let a = 1 and d = 2:

6 Strings Strings : Sequences of the form a 1, a 2,..., a n used in computer science are also called strings. This string is also denoted by a 1 a 2... a n. ( e.g. bit strings, which are finite sequences of bits) The length of a string is the number of terms in this string. The empty string, denoted by λ, is the string that has no terms. The empty string has length zero. Example: the string abcd is a string of length four.

7 Recurrence Relations Definition 4: A recurrence relation for the sequence {a n } is a equation that expresses a n in terms of one or more of the previous terms of the sequence, namely, a 0, a 1,, a n-1, for all integers n with n n 0, where n 0 is a nonnegative integer. A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. A recurrence relation is said to recursively define a sequence. The initial conditions for a sequence specify the terms that precede the first term where the recurrence relation takes effect.

8 Questions about Recurrence Relations Example 1: Let {a n } be a sequence that satisfies the recurrence relation a n = a n for n = 1,2,3,4,. and suppose that a 0 = 2. What are a 1, a 2 and a 3? [Here a 0 = 2 is the initial condition.] Solution: We see from the recurrence relation that a 1 = a = = 5 a 2 = = 8 a 3 = = 11

9 Questions about Recurrence Relations Example 2: Let {a n } be a sequence that satisfies the recurrence relation a n = a n-1 a n-2 for n = 2,3,4,. and suppose that a 0 = 3 and a 1 = 5. What are a 2,a 3 and a 4? [Here the initial conditions are a 0 = 3 and a 1 = 5. ] Solution: o We see from the recurrence relation that o a 2 = a 1 - a 0 = 5 3 = 2 o a 3 = a 2 a 1 = 2 5 = 3 o a 4 = a 3 a 3 = 3 2= 5

10 Fibonacci Sequence Definition 5: Define the Fibonacci sequence, f 0,f 1,f 2,, by: o Initial Conditions: f 0 = 0, f 1 = 1 o Recurrence Relation: f n = f n-1 + f n-2 for n=2,3,4. Example: Find f2,f3,f4, f5 and f6. Answer: o f2 = f1 + f0 = = 1, o f3 = f2 + f1 = = 2, o f4 = f3 + f2 = = 3, o f5 = f4 + f3 = = 5, o f6 = f5 + f4 = = 8.

11 Questions about Recurrence Relations Example : Determine whether the sequence {a n }, where a n = 3n for every nonnegative integer n, is a solution of the recurrence relation a n = 2 an 1 a n 2 for n = 2, 3, 4,.... Answer the same question where a n = 2 n and where a n = 5. Solution: o Suppose that a n = 3n for every nonnegative integer n. Then, for n 2, we see that 2a n 1 a n 2 = 2(3(n 1)) 3(n 2) = 3n = a n. Therefore, {a n }, where a n = 3n, is a solution of the recurrence relation. o Suppose that a n = 2 n for every nonnegative integer n. Note that a 0 = 1, a 1 = 2, and a 2 = 4. Because 2a 1 a 0 = = 3 a 2, we see that {a n }, where a n = 2 n, is not a solution of the recurrence relation. o Suppose that a n = 5 for every nonnegative integer n. Then for n 2, we see that a n = 2a n 1 a n 2 = = 5 = a n. Therefore, {a n }, where a n = 5, is a solution of the recurrence relation.

12 Solving Recurrence Relations Finding a formula for the nth term of the sequence generated by a recurrence relation together with the initial conditions is called solving the recurrence relation. Such a formula is called a closed formula.

13 Iterative Solution Example Method 1: Working upward, forward substitution Starting with the initial condition, and working upward until we reach a n to deduce a closed formula for the sequence. Example : Let {a n } be a sequence that satisfies the recurrence relation a n = a n for n = 2,3,4,. and suppose that a 1 = 2. a 2 = a 3 = (2 + 3) + 3 = a 4 = ( ) + 3 = a n = a n = (2 + 3 (n 2)) + 3 = 2 + 3(n 1)

14 Iterative Solution Example Method 2: Working downward, backward substitution starting with the term a n and working downward until we reach the initial condition to deduce this same formula. Example : Let {a n } be a sequence that satisfies the recurrence relation a n = a n for n = 2,3,4,. and suppose that a 1 = 2. a n = a n = (a n-2 + 3) + 3 = a n = (a n )+ 3 2 = a n = a 2 + 3(n 2) = (a 1 + 3) + 3(n 2) = 2 + 3(n 1)

15 Financial Application Example: Suppose that a person deposits $10, in a savings account at a bank yielding 11% per year with interest compounded annually. How much will be in the account after 30 years? Solution : o Let P n denote the amount in the account after 30 years. P n satisfies the following recurrence relation: o P n = P n P n-1 = (1.11) P n-1 with the initial condition P 0 = 10,000 o P 1 = (1.11)P 0 o P 2 = (1.11)P 1 = (1.11) 2 P 0 o P 3 = (1.11)P 2 = (1.11) 3 P 0 o : o P n = (1.11)P n-1 = (1.11) n P 0 = (1.11) n 10,000 o P n = (1.11) n 10,000 o P 30 = (1.11) 30 10,000 = $228,992.97

16 Special Integer Sequences Given a few terms of a sequence, try to identify the sequence. Conjecture a formula, recurrence relation, or some other rule. Some questions to ask? o Are there repeated terms of the same value? o Can you obtain a term from the previous term by adding an amount or multiplying by an amount? o Can you obtain a term by combining the previous terms in some way? o Are they cycles among the terms? o Do the terms match those of a well known sequence?

17 Questions on Special Integer Sequences Example 1: Find formulae for the sequences with the following first five terms: 1, 1/2, 1/4, 1/8, 1/16 Solution: Note that the denominators are powers of 2. The sequence with a n = 1/2 n is a possible match. This is a geometric progression with a = 1 and r = ½. Example 2: Consider 1,3,5,7,9 Solution: Note that each term is obtained by adding 2 to the previous term. A possible formula is a n = 2n + 1. This is an arithmetic progression with a =1 and d = 2.

18 Questions on Special Integer Sequences Example 3: How can we produce the terms of a sequence if the first 10 terms are 5, 11, 17, 23, 29, 35, 41, 47, 53, 59? Solution: Note that each of the first 10 terms of this sequence after the first is obtained by adding 6 to the previous term. Consequently, the nth term could be produced by starting with 5 and adding 6 a total of n 1 times; that is, a reasonable guess is that the nth term is 5 + 6(n 1) = 6n 1. This is an arithmetic progression with a = 5 and d = 6.

19 Useful Sequences

20 Questions on Special Integer Sequences Example : Conjecture a simple formula for an if the first 10 terms of the sequence {a n } are 1, 7, 25, 79, 241, 727, 2185, 6559, 19681, Solution: Comparing these terms with the corresponding terms of the sequence {3 n }, we notice that the nth term is 2 less than the corresponding power of 3. We see that a n = 3 n 2 for 1 n 10 and conjecture that this formula holds for all n.

21 Summations Sum of the terms from the sequence The notation: represents The variable j is called the index of summation. It runs through all the integers starting with its lower limit m and ending with its upper limit n.

22 Summations Example: What is the value of? Solution: Example : What is the value of? Solution :

23 Summations More generally for a set S: Example: What is the value of? Solution: o the sum of the values of s for all the members of the set {0, 2, 4},

24 Product Notation (optional) Product of the terms from the sequence The notation: represents

25 Geometric Series THEOREM 1: Sums of terms of geometric progressions

26 Double summations Double summations arise in many contexts (as in the analysis of nested loops in computer programs) o first expand the inner summation and then continue by computing the outer summation Example : Solution :

27

28 Section 2.6

29 Section Summary Definition of a Matrix Matrix Arithmetic Transposes and Powers of Arithmetic Zero-One matrices

30 Matrix Definition 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m n matrix. o The plural of matrix is matrices. o A matrix with the same number of rows as columns is called square. o Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal. 3 2 matrix

31 Notation Let m and n be positive integers and let The i th row of A is the 1 n matrix [a i1, a i2,,a in ]. The j th column of A is the m 1 matrix: The (i,j) th element or entry of A is the element a ij. We can use A = [a ij ] to denote the matrix with its (i,j) th element equal to a ij.

32 Matrix Arithmetic: Addition Definition 3: Let A = [a ij ] and B = [b ij ] be m n matrices. The sum of A and B, denoted by A + B, is the m n matrix that has a ij + b ij as its (i,j)th element. In other words, A + B = [a ij + b ij ]. Example: Note that matrices of different sizes can not be added.

33 Matrix Multiplication Definition 4: Let A be an m k matrix and B be a k n matrix. The product of A and B, denoted by A B, is the m n matrix that has its (i,j) th element equal to the sum of the products of the corresponding elments from the i th row of A and the j th column of B. In other words, if AB = [c ij ] then c ij = a i1 b 1j + a i2 b 2j + + a kj b 2j. Example: = The product of two matrices is undefined when the number of columns in the first matrix is not the same as the number of rows in the second.

34 Illustration of Matrix Multiplication The Product of A = [a ij ] and B = [b ij ]

35 Matrix Multiplication is not Commutative Example: Let Does AB = BA? Solution: AB BA

36 Identity Matrix and Powers of Matrices Definition 5: The identity matrix of order n is the n n matrix I n = [ ij ], where ij = 1 if i = j and ij = 0 if i j. when A is an m n matrix, AI n = I m A = A Powers of square matrices can be defined. When A is an n n matrix, we have: A 0 = I n A r = AAA A r times

37 Transposes of Matrices Definition 6: Let A = [a ij ] be an m n matrix. The transpose of A, denoted by A t,is the n m matrix obtained by interchanging the rows and columns of A. If A t = [b ij ], then b ij = a ji for i =1,2,,n and j = 1,2,...,m.

38 Transposes of Matrices Definition 7: A square matrix A is called symmetric if A = A t. Thus A = [a ij ] is symmetric if a ij = a ji for i and j with 1 i n and 1 j n. Square matrices do not change when their rows and columns are interchanged.

39 Zero-One Matrices A matrix all of whose entries are either 0 or 1 is called a zeroone matrix. Algorithms operating on discrete structures represented by zero-one matrices are based on Boolean arithmetic defined by the following Boolean operations:

40 Zero-One Matrices Definition 8: Let A = [a ij ] and B = [b ij ] be an m n zero-one matrices. o The join of A and B is the zero-one matrix with (i,j)th entry a ij b ij. The join of A and B is denoted by A B. o The meet of of A and B is the zero-one matrix with (i,j)th entry a ij b ij. The meet of A and B is denoted by A B.

41 Joins and Meets of Zero-One Matrices Example: Find the join and meet of the zero-one matrices Solution: o The join of A and B is o The meet of A and B is

42 Boolean Product of Zero-One Matrices Definition 9: Let A = [a ij ] be an m k zero-one matrix and B = [b ij ] be a k n zero-one matrix. The Boolean product of A and B, denoted by A B, is the m n zero-one matrix with(i,j)th entry c ij = (a i1 b 1j ) (a i2 b 2j ) (a ik b kj ). Example: Find the Boolean product of A and B, where

43 Boolean Powers of Zero-One Matrices Definition 10: Let A be a square zero-one matrix and let r be a positive integer. The rth Boolean power of A is the Boolean product of r factors of A, denoted by A [r]. Hence, We define A [0] to be In. The Boolean product is well defined because the Boolean product of matrices is associative.

44 Boolean Powers of Zero-One Matrices Example: Let Find A n for all positive integers n. Solution:

Section Summary. Sequences. Recurrence Relations. Summations. Examples: Geometric Progression, Arithmetic Progression. Example: Fibonacci Sequence

Section Summary. Sequences. Recurrence Relations. Summations. Examples: Geometric Progression, Arithmetic Progression. Example: Fibonacci Sequence Section 2.4 1 Section Summary Sequences. Examples: Geometric Progression, Arithmetic Progression Recurrence Relations Example: Fibonacci Sequence Summations 2 Introduction Sequences are ordered lists of

More information

Section Summary. Sequences. Recurrence Relations. Summations. Examples: Geometric Progression, Arithmetic Progression. Example: Fibonacci Sequence

Section Summary. Sequences. Recurrence Relations. Summations. Examples: Geometric Progression, Arithmetic Progression. Example: Fibonacci Sequence Section 2.4 Section Summary Sequences. Examples: Geometric Progression, Arithmetic Progression Recurrence Relations Example: Fibonacci Sequence Summations Introduction Sequences are ordered lists of elements.

More information

Section Summary. Definition of a Function.

Section Summary. Definition of a Function. Section 2.3 Section Summary Definition of a Function. Domain, Codomain Image, Preimage Injection, Surjection, Bijection Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial

More information

Sec$on Summary. Sequences. Recurrence Relations. Summations. Ex: Geometric Progression, Arithmetic Progression. Ex: Fibonacci Sequence

Sec$on Summary. Sequences. Recurrence Relations. Summations. Ex: Geometric Progression, Arithmetic Progression. Ex: Fibonacci Sequence Section 2.4 Sec$on Summary Sequences Ex: Geometric Progression, Arithmetic Progression Recurrence Relations Ex: Fibonacci Sequence Summations 2 Introduc$on Sequences are ordered lists of elements. 1, 2,

More information

Definition: A sequence is a function from a subset of the integers (usually either the set

Definition: A sequence is a function from a subset of the integers (usually either the set Math 3336 Section 2.4 Sequences and Summations Sequences Geometric Progression Arithmetic Progression Recurrence Relation Fibonacci Sequence Summations Definition: A sequence is a function from a subset

More information

Discrete Structures Lecture Sequences and Summations

Discrete Structures Lecture Sequences and Summations Introduction Good morning. In this section we study sequences. A sequence is an ordered list of elements. Sequences are important to computing because of the iterative nature of computer programs. The

More information

Countable and uncountable sets. Matrices.

Countable and uncountable sets. Matrices. CS 441 Discrete Mathematics for CS Lecture 11 Countable and uncountable sets. Matrices. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Arithmetic series Definition: The sum of the terms of the

More information

Countable and uncountable sets. Matrices.

Countable and uncountable sets. Matrices. Lecture 11 Countable and uncountable sets. Matrices. Instructor: Kangil Kim (CSE) E-mail: kikim01@konkuk.ac.kr Tel. : 02-450-3493 Room : New Milenium Bldg. 1103 Lab : New Engineering Bldg. 1202 Next topic:

More information

CS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:

CS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages: CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246-262 Matrices 2 Introduction DEFINITION 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n

More information

Sets are one of the basic building blocks for the types of objects considered in discrete mathematics.

Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. Section 2.1 Introduction Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. Important for counting. Programming languages have set operations. Set theory

More information

Chapter 2 - Basics Structures MATH 213. Chapter 2: Basic Structures. Dr. Eric Bancroft. Fall Dr. Eric Bancroft MATH 213 Fall / 60

Chapter 2 - Basics Structures MATH 213. Chapter 2: Basic Structures. Dr. Eric Bancroft. Fall Dr. Eric Bancroft MATH 213 Fall / 60 MATH 213 Chapter 2: Basic Structures Dr. Eric Bancroft Fall 2013 Dr. Eric Bancroft MATH 213 Fall 2013 1 / 60 Chapter 2 - Basics Structures 2.1 - Sets 2.2 - Set Operations 2.3 - Functions 2.4 - Sequences

More information

Chapter 2 - Basics Structures

Chapter 2 - Basics Structures Chapter 2 - Basics Structures 2.1 - Sets Definitions and Notation Definition 1 (Set). A set is an of. These are called the or of the set. We ll typically use uppercase letters to denote sets: S, A, B,...

More information

Matrices. Chapter Definitions and Notations

Matrices. Chapter Definitions and Notations Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which

More information

Matrix Multiplication

Matrix Multiplication 3.2 Matrix Algebra Matrix Multiplication Example Foxboro Stadium has three main concession stands, located behind the south, north and west stands. The top-selling items are peanuts, hot dogs and soda.

More information

Elementary maths for GMT

Elementary maths for GMT Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1

More information

Recurrence Relations

Recurrence Relations Recurrence Relations Recurrence Relations Reading (Epp s textbook) 5.6 5.8 1 Recurrence Relations A recurrence relation for a sequence aa 0, aa 1, aa 2, ({a n }) is a formula that relates each term a k

More information

Phys 201. Matrices and Determinants

Phys 201. Matrices and Determinants Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1

More information

Section 9.2: Matrices.. a m1 a m2 a mn

Section 9.2: Matrices.. a m1 a m2 a mn Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in

More information

Matrices: 2.1 Operations with Matrices

Matrices: 2.1 Operations with Matrices Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION x 1,, x n A linear equation in the variables equation that can be written in the form a 1 x 1 + a 2 x 2 + + a n x n

More information

Lecture 3 Linear Algebra Background

Lecture 3 Linear Algebra Background Lecture 3 Linear Algebra Background Dan Sheldon September 17, 2012 Motivation Preview of next class: y (1) w 0 + w 1 x (1) 1 + w 2 x (1) 2 +... + w d x (1) d y (2) w 0 + w 1 x (2) 1 + w 2 x (2) 2 +...

More information

Matrices BUSINESS MATHEMATICS

Matrices BUSINESS MATHEMATICS Matrices BUSINESS MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question

More information

Matrix Arithmetic. j=1

Matrix Arithmetic. j=1 An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the i-th row and j-th column

More information

Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.

Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2

More information

Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds

Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds These notes are meant to provide a brief introduction to the topics from Linear Algebra that will be useful in Math3315/CSE3365, Introduction

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

MATRICES. a m,1 a m,n A =

MATRICES. a m,1 a m,n A = MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of

More information

Sequences A sequence is a function, where the domain is a set of consecutive positive integers beginning with 1.

Sequences A sequence is a function, where the domain is a set of consecutive positive integers beginning with 1. 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 8: Sequences, Series, and Combinatorics Section 8.1 Sequences and Series Sequences A sequence

More information

MATH2210 Notebook 2 Spring 2018

MATH2210 Notebook 2 Spring 2018 MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I ICS4: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Jan Stelovsky based on slides by Dr. Baek and Dr. Still Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by

More information

Review of Vectors and Matrices

Review of Vectors and Matrices A P P E N D I X D Review of Vectors and Matrices D. VECTORS D.. Definition of a Vector Let p, p, Á, p n be any n real numbers and P an ordered set of these real numbers that is, P = p, p, Á, p n Then P

More information

. a m1 a mn. a 1 a 2 a = a n

. a m1 a mn. a 1 a 2 a = a n Biostat 140655, 2008: Matrix Algebra Review 1 Definition: An m n matrix, A m n, is a rectangular array of real numbers with m rows and n columns Element in the i th row and the j th column is denoted by

More information

Matrices and Vectors

Matrices and Vectors Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix

More information

Two matrices of the same size are added by adding their corresponding entries =.

Two matrices of the same size are added by adding their corresponding entries =. 2 Matrix algebra 2.1 Addition and scalar multiplication Two matrices of the same size are added by adding their corresponding entries. For instance, 1 2 3 2 5 6 3 7 9 +. 4 0 9 4 1 3 0 1 6 Addition of two

More information

Lecture 3: Matrix and Matrix Operations

Lecture 3: Matrix and Matrix Operations Lecture 3: Matrix and Matrix Operations Representation, row vector, column vector, element of a matrix. Examples of matrix representations Tables and spreadsheets Scalar-Matrix operation: Scaling a matrix

More information

Chapter 5: Sequences, Mathematic Induction, and Recursion

Chapter 5: Sequences, Mathematic Induction, and Recursion Chapter 5: Sequences, Mathematic Induction, and Recursion Hao Zheng Department of Computer Science and Engineering University of South Florida Tampa, FL 33620 Email: zheng@cse.usf.edu Phone: (813)974-4757

More information

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M. S. KumarSwamy, TGT(Maths) Page Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 50 - CHAPTER 3: MATRICES QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 03 marks Matrix A matrix is an ordered rectangular array of numbers

More information

Linear Algebra V = T = ( 4 3 ).

Linear Algebra V = T = ( 4 3 ). Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional

More information

Introduction. Vectors and Matrices. Vectors [1] Vectors [2]

Introduction. Vectors and Matrices. Vectors [1] Vectors [2] Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector - one dimensional array Matrix -

More information

A Review of Matrix Analysis

A Review of Matrix Analysis Matrix Notation Part Matrix Operations Matrices are simply rectangular arrays of quantities Each quantity in the array is called an element of the matrix and an element can be either a numerical value

More information

Mathematics 13: Lecture 10

Mathematics 13: Lecture 10 Mathematics 13: Lecture 10 Matrices Dan Sloughter Furman University January 25, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, 2008 1 / 19 Matrices Recall: A matrix is a

More information

Recurrence Relations and Recursion: MATH 180

Recurrence Relations and Recursion: MATH 180 Recurrence Relations and Recursion: MATH 180 1: Recursively Defined Sequences Example 1: The sequence a 1,a 2,a 3,... can be defined recursively as follows: (1) For all integers k 2, a k = a k 1 + 1 (2)

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

8. Sequences, Series, and Probability 8.1. SEQUENCES AND SERIES

8. Sequences, Series, and Probability 8.1. SEQUENCES AND SERIES 8. Sequences, Series, and Probability 8.1. SEQUENCES AND SERIES What You Should Learn Use sequence notation to write the terms of sequences. Use factorial notation. Use summation notation to write sums.

More information

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Matrix Arithmetic Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)

More information

COM S 330 Lecture Notes Week of Feb 9 13

COM S 330 Lecture Notes Week of Feb 9 13 Monday, February 9. Rosen.4 Sequences Reading: Rosen.4. LLM 4.. Ducks 8., 8., Def: A sequence is a function from a (usually infinite) subset of the integers (usually N = {0,,, 3,... } or Z + = {,, 3, 4,...

More information

CLASS 12 ALGEBRA OF MATRICES

CLASS 12 ALGEBRA OF MATRICES CLASS 12 ALGEBRA OF MATRICES Deepak Sir 9811291604 SHRI SAI MASTERS TUITION CENTER CLASS 12 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements

More information

1 Matrices and Systems of Linear Equations. a 1n a 2n

1 Matrices and Systems of Linear Equations. a 1n a 2n March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real

More information

Sequences and Series. College Algebra

Sequences and Series. College Algebra Sequences and Series College Algebra Sequences A sequence is a function whose domain is the set of positive integers. A finite sequence is a sequence whose domain consists of only the first n positive

More information

Matrix representation of a linear map

Matrix representation of a linear map Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors

More information

3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions

3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence

More information

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections ) c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side

More information

Matrix representation of a linear map

Matrix representation of a linear map Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors

More information

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x

More information

Linear Algebra March 16, 2019

Linear Algebra March 16, 2019 Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented

More information

Stage-structured Populations

Stage-structured Populations Department of Biology New Mexico State University Las Cruces, New Mexico 88003 brook@nmsu.edu Fall 2009 Age-Structured Populations All individuals are not equivalent to each other Rates of survivorship

More information

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations MTHSC 3110 Section 2.1 Matrix Operations Notation Let A be an m n matrix, that is, m rows and n columns. We ll refer to the entries of A by their row and column indices. The entry in the i th row and j

More information

DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.

DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra. DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1

More information

Chapter 2. Ma 322 Fall Ma 322. Sept 23-27

Chapter 2. Ma 322 Fall Ma 322. Sept 23-27 Chapter 2 Ma 322 Fall 2013 Ma 322 Sept 23-27 Summary ˆ Matrices and their Operations. ˆ Special matrices: Zero, Square, Identity. ˆ Elementary Matrices, Permutation Matrices. ˆ Voodoo Principle. What is

More information

Announcements Wednesday, October 10

Announcements Wednesday, October 10 Announcements Wednesday, October 10 The second midterm is on Friday, October 19 That is one week from this Friday The exam covers 35, 36, 37, 39, 41, 42, 43, 44 (through today s material) WeBWorK 42, 43

More information

Functions. Given a function f: A B:

Functions. Given a function f: A B: Functions Given a function f: A B: We say f maps A to B or f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f.

More information

Matrices. 1 a a2 1 b b 2 1 c c π

Matrices. 1 a a2 1 b b 2 1 c c π Matrices 2-3-207 A matrix is a rectangular array of numbers: 2 π 4 37 42 0 3 a a2 b b 2 c c 2 Actually, the entries can be more general than numbers, but you can think of the entries as numbers to start

More information

Matrix Basic Concepts

Matrix Basic Concepts Matrix Basic Concepts Topics: What is a matrix? Matrix terminology Elements or entries Diagonal entries Address/location of entries Rows and columns Size of a matrix A column matrix; vectors Special types

More information

Section-A. Short Questions

Section-A. Short Questions Section-A Short Questions Question1: Define Problem? : A Problem is defined as a cultural artifact, which is especially visible in a society s economic and industrial decision making process. Those managers

More information

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 11 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 real

More information

CSCE 222 Discrete Structures for Computing. Dr. Hyunyoung Lee

CSCE 222 Discrete Structures for Computing. Dr. Hyunyoung Lee CSCE 222 Discrete Structures for Computing Sequences and Summations Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Sequences 2 Sequences A sequence is a function from a subset of the set of

More information

Math 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction

Math 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If

More information

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column

More information

Chapter 8: Recursion. March 10, 2008

Chapter 8: Recursion. March 10, 2008 Chapter 8: Recursion March 10, 2008 Outline 1 8.1 Recursively Defined Sequences 2 8.2 Solving Recurrence Relations by Iteration 3 8.4 General Recursive Definitions Recursively Defined Sequences As mentioned

More information

CS Discrete Mathematics Dr. D. Manivannan (Mani)

CS Discrete Mathematics Dr. D. Manivannan (Mani) CS 275 - Discrete Mathematics Dr. D. Manivannan (Mani) Department of Computer Science University of Kentucky Lexington, KY 40506 Course Website: www.cs.uky.edu/~manivann/cs275 Notes based on Discrete Mathematics

More information

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) ( ) Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

More information

ICS 6N Computational Linear Algebra Matrix Algebra

ICS 6N Computational Linear Algebra Matrix Algebra ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine xhx@uci.edu February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, 2017 1 / 24 Matrix Consider an m n matrix

More information

. =. a i1 x 1 + a i2 x 2 + a in x n = b i. a 11 a 12 a 1n a 21 a 22 a 1n. i1 a i2 a in

. =. a i1 x 1 + a i2 x 2 + a in x n = b i. a 11 a 12 a 1n a 21 a 22 a 1n. i1 a i2 a in Vectors and Matrices Continued Remember that our goal is to write a system of algebraic equations as a matrix equation. Suppose we have the n linear algebraic equations a x + a 2 x 2 + a n x n = b a 2

More information

Systems of Linear Equations and Matrices

Systems of Linear Equations and Matrices Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2 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

More information

Elementary Row Operations on Matrices

Elementary Row Operations on Matrices King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix

More information

Sequences are ordered lists of elements

Sequences are ordered lists of elements Sequences are ordered lists of elements Definition: A sequence is a function from the set of integers, either set {0,1,2,3, } or set {1,2,3,4,..}, to a set S. We use the notation a n to denote the image

More information

Matrix Algebra & Elementary Matrices

Matrix Algebra & Elementary Matrices Matrix lgebra & Elementary Matrices To add two matrices, they must have identical dimensions. To multiply them the number of columns of the first must equal the number of rows of the second. The laws below

More information

Systems of Linear Equations and Matrices

Systems of Linear Equations and Matrices Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

More information

Matrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices

Matrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated

More information

Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX

Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX September 2007 MSc Sep Intro QT 1 Who are these course for? The September

More information

Linear System of Equations

Linear System of Equations Linear System of Equations Linear systems are perhaps the most widely applied numerical procedures when real-world situation are to be simulated. Example: computing the forces in a TRUSS. F F 5. 77F F.

More information

Propositional Logic, Predicates, and Equivalence

Propositional Logic, Predicates, and Equivalence Chapter 1 Propositional Logic, Predicates, and Equivalence A statement or a proposition is a sentence that is true (T) or false (F) but not both. The symbol denotes not, denotes and, and denotes or. If

More information

12 Sequences and Recurrences

12 Sequences and Recurrences 12 Sequences and Recurrences A sequence is just what you think it is. It is often given by a formula known as a recurrence equation. 12.1 Arithmetic and Geometric Progressions An arithmetic progression

More information

Matrices and Linear Algebra

Matrices and Linear Algebra Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2

More information

Think about systems of linear equations, their solutions, and how they might be represented with matrices.

Think about systems of linear equations, their solutions, and how they might be represented with matrices. Think About This Situation Unit 4 Lesson 3 Investigation 1 Name: Think about systems of linear equations, their solutions, and how they might be represented with matrices. a Consider a system of two linear

More information

A primer on matrices

A primer on matrices A primer on matrices Stephen Boyd August 4, 2007 These notes describe the notation of matrices, the mechanics of matrix manipulation, and how to use matrices to formulate and solve sets of simultaneous

More information

MATH Mathematics for Agriculture II

MATH Mathematics for Agriculture II MATH 10240 Mathematics for Agriculture II Academic year 2018 2019 UCD School of Mathematics and Statistics Contents Chapter 1. Linear Algebra 1 1. Introduction to Matrices 1 2. Matrix Multiplication 3

More information

Jim Lambers MAT 610 Summer Session Lecture 1 Notes

Jim Lambers MAT 610 Summer Session Lecture 1 Notes Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra

More information

CHAPTER 6. Direct Methods for Solving Linear Systems

CHAPTER 6. Direct Methods for Solving Linear Systems CHAPTER 6 Direct Methods for Solving Linear Systems. Introduction A direct method for approximating the solution of a system of n linear equations in n unknowns is one that gives the exact solution to

More information

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices Graphics 2009/2010, period 1 Lecture 4 Matrices m n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in

More information

Linear Algebra and Matrix Inversion

Linear Algebra and Matrix Inversion Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much

More information

Section 5.5: Matrices and Matrix Operations

Section 5.5: Matrices and Matrix Operations Section 5.5 Matrices and Matrix Operations 359 Section 5.5: Matrices and Matrix Operations Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.

More information

Matrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.

Matrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via. Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n

More information

Fundamentals of Engineering Analysis (650163)

Fundamentals of Engineering Analysis (650163) Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is

More information

CHAPTER 7: Systems and Inequalities

CHAPTER 7: Systems and Inequalities (Exercises for Chapter 7: Systems and Inequalities) E.7.1 CHAPTER 7: Systems and Inequalities (A) means refer to Part A, (B) means refer to Part B, etc. (Calculator) means use a calculator. Otherwise,

More information

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ] Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.

More information