# Quiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown.

Save this PDF as:

Size: px
Start display at page:

Download "Quiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown."

## Transcription

1 Quiz 1) Simplify ) Locate your 1 st order neighbors Name Hometown Me Name Hometown Name Hometown Name Hometown

2 Solving Linear Algebraic Equa3ons

3 Basic Concepts Here only real vector and matrices are considered. Column vector Row vector Scalar product Norm (length)

4 Basic Concepts Matrix of order Matrix A is square if m = n Matrix A consists of columns of vectors a j

5 Basic Concepts Transpose of A: (becomes matrix of order n m) Diagonal matrix:

6 Basic Concepts Identity matrix Inverse of matrix A is A -1 satisfying A is symmetric if A is orthogonal if A is square and

7 Basic Concepts Lower triangular matrix Upper triangular matrix goes analogously Exponential (A is square matrix, I is identity matrix)

8 (1) Addition Matrix Algebra (2) Subtraction For (1) and (2), A, B, and C of the same order (3) Multiplication by a number (4) Product of two matrices (in general)

9 Exponential relations Matrix Algebra

10 Trace If A is square matrix of order n n, then Trace is the sum of the diagonal elements. for order The trace of a matrix A is the sum of its eigenvalues

11 Determinants A determinant is a real number associated with every square matrix. The determinant of a square matrix A is denoted by "det A" or A. The Leibniz formula for the determinant of an n n matrix A is

12 Determinants If A is a square matrix, then the minor of the entry in the i-th row and j-th column is the determinant of the sub-matrix formed by deleting the i-th row and j-th column. The (i,j) cofactor is obtained by multiplying the minor by (-1) i+j.

13 Determinants For n = 2: For n = 3 (example, operation along the 1 st row):

14 Determinants For n = 4 (example, operation along the 2 nd column):

15 Triangular determinant Determinants

16 Determinants (7) If all elements of a row (or a column) are multiplied by a constant c, the determinant is multiplied by c. (8) Exchange of two rows (or columns) changes the sign of the determinant.

17 Determinants (9) A determinant does not change if one row (column) multiplied by a constant is added to another row (column). (10) A determinant equals zero if (i) all elements of a row (column) are zero, or (ii) two rows (columns) coincide. (11) Sub-determinant D ij of order (n-1) (n-1) is formed by deleting the i th row and the j th column of D. Cofactor

18 Matrix Inversion The inverse of a square matrix A (also called reciprocal matrix), is a matrix A 1, such that where I is the identity matrix. If matrix A has an inverse, then it is called nonsingular or, also invertible. If linearly independent. columns (rows) are

19 Matrix Inversion There are many methods to calculate A 1 : (1) Cofactor method where C is the matrix cofactors. So

20 Matrix Inversion (2) Elementary row operations from which (3) Newton s method (4) Eigendecomposition (5) Cholesky decomposition (6) etc

21 Inversion of 2 2 matrices Inversion of 3 3 matrices

22 MATLAB Implementa3ons MATLAB has built-in functions/commands for matrix operations: >> det(a) - computes matrix determinant >> inv(a) - computes matrix inverse >> trace(a) - computes matrix trace >> eig(a) - computes eigenvalues >> rank(a) - computes matrix rank >> eye(a) - generates identity matrix >> diag(a) - returns matrix diagonal elements >> sum(diag(a)) - computes matrix trace etc.

23 MATLAB Implementa3ons Creating a simple solver/function that calculates the determinant of 3 3 matrices function [d] = Det3by3(a) % Function to calculate determinant matrices size 3x3 % Operation based on the 1st row row1 = a(1,:); dd1 = a(2,2)*a(3,3) - a(2,3)*a(3,2); dd2 = a(2,1)*a(3,3) - a(2,3)*a(3,1); dd3 = a(2,1)*a(3,2) - a(2,2)*a(3,1); d = (-1)^2*row1(1,1)*dd1 + (-1)^3*row1(1,2)*dd (-1)^4*row1(1,3)*dd3;

24 MATLAB Implementa3ons To use the solver, on the Command Window: >> u=[1,1,0;4,5,9;0,2,8] u = >> Det3by3(u) ans -10

25 Systems of Linear Equa3ons The general form of systems of linear equations (3.1) where a s are constant coefficients, b s are constants, m is the number of equations, and n is the number of variables or unknowns. The system (3.1) is homogeneous if all otherwise inhomogeneous.

26 Systems of Linear Equa3ons Arranging in matrix and vector notation or (3.2) The matrix A is called the coefficient matrix and has size (m rows and n columns).

27 Systems of Linear Equa3ons Solutions of general linear systems with m equations in n unknowns may be classified into exactly three possibilities: (1) No solution (2) Infinitely many solutions (3) A unique solution

28 Solving Equa3ons with Geometry If n = 2 (plane) or n = 3 (space), the system of equations (3.1) has a geometric interpretation that can give us valuable intuition about possible solutions. Plane Geometry - represented by line equations (3.3) Solving the system above is the geometrical equivalent of finding all possible (x,y)-intersections of the lines.

29 Solving Equa3ons with Geometry Parallel lines, no solution Identical lines, infinitely many solutions Non-parallel distinct lines, one solution at the unique intersection

30 Solving Equa3ons with Geometry Space Geometry - A plane in xyz-space is given by plane equation The vector is normal to the plane. (3.4) Solving the system above is the geometrical equivalent of finding all possible (x,y,z)-intersections of the planes.

31 Solving Equa3ons with Geometry A B C A=B Triple-decker. Planes A, B, and C are parallel. No intersection point. Double-decker. Planes A and B are equal and parallel to plane C. No intersection. C C A B Book shelf. Planes A and B are distinct and parallel. No intersection point.

32

33 Solving Equa3ons with Matrix Op. A system of m linear equations and n unknowns can be written in matrix notation as: or solving it: (3.5) (3.6)

34 Solving Equa3ons with Matrix Op. If m=n in Eq. (3.1) and if det A 0, the unique solution can be expressed explicitly by either: where det A is the determinant of matrix A, det A j is the determinant that arises when the j-th column of det A is replaced by the column elements b 1,, b n (Cramer s rule).

35 MATLAB Implementa3ons Solve the following system of equations using MATLAB: >> A = [0.3,0.52,1;0.5,1,1.9;0.1,0.3,0.5]; >> b = [-0.01, 0.67, -0.44] ; >> x = A\b x =

36 Solving of Equa3ons by Gaussian Elimina3on If m n in Eq. (3.1), Gaussian elimination can be employed to solve the system of equations (3.1). By elementary row operations (i.e., (i) exchange of equations, (ii) multiplication of an equation by a constant, (iii) adding one equation to another), the system (3.1) can be transformed into echelon form. From the echelon form, the unknowns n are determined by backward substitution. Of these there are 2 groups: 1) Basic variables: corresponding to pivots 2) Free variables: the others

37 Solving of Equa3ons by Example: Gaussian Elimina3on x,y,z are basic variables, u is free variable.

38 Eigenvalues The number λ is called an eigenvalue of a square matrix A and g 0 is a corresponding eigenvector if (3.7) Characteristic equation is the equation which is solved to find a matrix s eigenvalues. The characteristic equation in variable λ is defined by

39 Eigenvalues (product of all eigenvalues) (sum of eigenvalues) (Real square) matrix A is symmetric if If matrix A is symmetric: (1) all eigenvalues are real (2) eigenvectors corresponding to different eigenvalues are orthogonal (3)

40 Eigenvalues The characteristic equation of a 2x2 matrix: or can be written in the particular form: Similarly, the characteristic equation of a 3x3 matrix:

41 Eigenvalues or, can be written as well as where tr(a) is the matrix trace of A and det(a) is its determinant.

42 Spectral Theorem Assume that: (1) Matrix A is symmetric (2) are its eigenvalues Then there exist corresponding eigenvectors g 1,, g n such that: (1) (pairwise orthogonal and normed) Set, then (2) P is an orthogonal matrix (3) (4) (5)

43 Eigenvalues & Eigenvectors Example: Find eigenvalues and eigenvectors of matrix Solution The characteristic equation of matrix A is with roots

44 Eigenvalues & Eigenvectors Using Eq. (3.3), which states and we get If we choose the general solutions

45 Eigenvalues & Eigenvectors then the unknown vector is given by Since eigenvectors are normalized, then From

46 Eigenvalues & Eigenvectors we get If we choose the general solutions then the second unknown vector is given by

47 Eigenvalues & Eigenvectors From Choosing the general solutions the third eigenvector is given by

48 MATLAB Implementa3ons Matlab commands for matrix operations: >> A=[8,1,6;3,5,7;4,9,2] >> B = A >> trace(a) >> rank(a) >> det(a) >> inv(a)

49 MATLAB Implementa3ons Creating a diagonal matrix from a vector: >> A=[1,2,4,5,7,9]; >> B=diag(A) B =

50 MATLAB Implementa3ons Generating a vector: >> A=zeros(1,2) A = 0 0 Transpose of a vector: >> B=A B = 0 0 Creating a 2 3 matrix: >> D=[1,2,4;5,7,9] D = Multiplication of matrices/vectors: >> C=A.*B; Suppress output

51 MATLAB Implementa3ons Generating a 5 5 matrix whose elements are all zeros: Generating a 5 3 matrix whose elements are all ones >> A=zeros(5) A = >> B=ones(5,3) B =

52 MATLAB Implementa3ons Concatenating matrices A and B horizontally : >> C=[A,B] C = (Same length of rows) Concatenating matrices vertically: >> C=[A;B] (Same length of columns)

### 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

### 1 Determinants. 1.1 Determinant

1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to

### MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.

MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of

### Examples and MatLab. Vector and Matrix Material. Matrix Addition R = A + B. Matrix Equality A = B. Matrix Multiplication R = A * B.

Vector and Matrix Material Examples and MatLab Matrix = Rectangular array of numbers, complex numbers or functions If r rows & c columns, r*c elements, r*c is order of matrix. A is n * m matrix Square

### 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

### CHAPTER 8: Matrices and Determinants

(Exercises for Chapter 8: Matrices and Determinants) E.8.1 CHAPTER 8: Matrices and Determinants (A) means refer to Part A, (B) means refer to Part B, etc. Most of these exercises can be done without a

### ANSWERS. E k E 2 E 1 A = B

MATH 7- Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij

Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix

### Topic 15 Notes Jeremy Orloff

Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.

### 1 9/5 Matrices, vectors, and their applications

1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric

### c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.

### Contents. 1 Vectors, Lines and Planes 1. 2 Gaussian Elimination Matrices Vector Spaces and Subspaces 124

Matrices Math 220 Copyright 2016 Pinaki Das This document is freely redistributable under the terms of the GNU Free Documentation License For more information, visit http://wwwgnuorg/copyleft/fdlhtml Contents

### and let s calculate the image of some vectors under the transformation T.

Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### Matrix & Linear Algebra

Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A

### Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant

### Linear Algebra Primer

Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................

### Methods for Solving Linear Systems Part 2

Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use

### HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)

HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe

### Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion

Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates

### Math Linear Algebra Final Exam Review Sheet

Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

### Math 215 HW #9 Solutions

Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

### (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

### 4. Determinants.

4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.

### A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

### Introduction to Matrix Algebra

Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

### MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra

MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;

### EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2

EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko

### Chapter 7. Linear Algebra: Matrices, Vectors,

Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.

### SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to 1.1. Introduction Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that

### Calculating determinants for larger matrices

Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

### Chapter 4. Determinants

4.2 The Determinant of a Square Matrix 1 Chapter 4. Determinants 4.2 The Determinant of a Square Matrix Note. In this section we define the determinant of an n n matrix. We will do so recursively by defining

### Linear Algebra in Actuarial Science: Slides to the lecture

Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations

Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications to the natural

### TOPIC III LINEAR ALGEBRA

[1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:

### Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system

### 18.06SC Final Exam Solutions

18.06SC Final Exam Solutions 1 (4+7=11 pts.) Suppose A is 3 by 4, and Ax = 0 has exactly 2 special solutions: 1 2 x 1 = 1 and x 2 = 1 1 0 0 1 (a) Remembering that A is 3 by 4, find its row reduced echelon

### Basic Concepts in Linear Algebra

Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear

### Math 240 Calculus III

The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A

### APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF

ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done

### Matrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A =

30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = a 11 a 12... a 1N a 21 a 22... a 2N...... a M1 a M2... a MN A matrix can

### Extra Problems for Math 2050 Linear Algebra I

Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as

### Lecture 10: Determinants and Cramer s Rule

Lecture 0: Determinants and Cramer s Rule The determinant and its applications. Definition The determinant of a square matrix A, denoted by det(a) or A, is a real number, which is defined as follows. -by-

### MATH 1210 Assignment 4 Solutions 16R-T1

MATH 1210 Assignment 4 Solutions 16R-T1 Attempt all questions and show all your work. Due November 13, 2015. 1. Prove using mathematical induction that for any n 2, and collection of n m m matrices A 1,

### 1. General Vector Spaces

1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

### 1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )

Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical

### Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University

Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,

### Determinants Chapter 3 of Lay

Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j

### Numerical Linear Algebra

Numerical Linear Algebra Direct Methods Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) Linear Systems: Direct Solution Methods Fall 2017 1 / 14 Introduction The solution of linear systems is one

### MTH 102A - Linear Algebra II Semester

MTH 0A - Linear Algebra - 05-6-II Semester Arbind Kumar Lal P Field A field F is a set from which we choose our coefficients and scalars Expected properties are ) a+b and a b should be defined in it )

### Matrix Operations: Determinant

Matrix Operations: Determinant Determinants Determinants are only applicable for square matrices. Determinant of the square matrix A is denoted as: det(a) or A Recall that the absolute value of the determinant

### Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

### 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 -

### SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra

1.1. Introduction SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that

### Basic Concepts in Matrix Algebra

Basic Concepts in Matrix Algebra An column array of p elements is called a vector of dimension p and is written as x p 1 = x 1 x 2. x p. The transpose of the column vector x p 1 is row vector x = [x 1

### Eigenvalues and Eigenvectors

Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Eigenvalues are the key to a system of n differential equations : dy=dt ay becomes dy=dt Ay. Now A is a matrix and y is a vector.y.t/;

### Introduction to Determinants

Introduction to Determinants For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix The matrix A is invertible if and only if.

### Numerical Methods I Solving Square Linear Systems: GEM and LU factorization

Numerical Methods I Solving Square Linear Systems: GEM and LU factorization Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 September 18th,

### MA 265 FINAL EXAM Fall 2012

MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators

### Introduction to Matrices

214 Analysis and Design of Feedback Control Systems Introduction to Matrices Derek Rowell October 2002 Modern system dynamics is based upon a matrix representation of the dynamic equations governing the

### Linear Algebra for Machine Learning. Sargur N. Srihari

Linear Algebra for Machine Learning Sargur N. srihari@cedar.buffalo.edu 1 Overview Linear Algebra is based on continuous math rather than discrete math Computer scientists have little experience with it

### a11 a A = : a 21 a 22

Matrices The study of linear systems is facilitated by introducing matrices. Matrix theory provides a convenient language and notation to express many of the ideas concisely, and complicated formulas are

### Notes on Mathematics

Notes on Mathematics - 12 1 Peeyush Chandra, A. K. Lal, V. Raghavendra, G. Santhanam 1 Supported by a grant from MHRD 2 Contents I Linear Algebra 7 1 Matrices 9 1.1 Definition of a Matrix......................................

### Math Camp Notes: Linear Algebra I

Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n

### MATH 369 Linear Algebra

Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine

### Linear Algebra: Characteristic Value Problem

Linear Algebra: Characteristic Value Problem . The Characteristic Value Problem Let < be the set of real numbers and { be the set of complex numbers. Given an n n real matrix A; does there exist a number

### Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C

Topic 1 Quiz 1 text A reduced row-echelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correct-choice may have 0, 1, 2, or 3 choice may have 0,

### Mathematical Foundations

Chapter 1 Mathematical Foundations 1.1 Big-O Notations In the description of algorithmic complexity, we often have to use the order notations, often in terms of big O and small o. Loosely speaking, for

### Chapter 2: Matrices and Linear Systems

Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers

### 1. Select the unique answer (choice) for each problem. Write only the answer.

MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

### MATH 310, REVIEW SHEET 2

MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,

### 18.06 Problem Set 8 - Solutions Due Wednesday, 14 November 2007 at 4 pm in

806 Problem Set 8 - Solutions Due Wednesday, 4 November 2007 at 4 pm in 2-06 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state

### 1 Linearity and Linear Systems

Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 7-8 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)

### Quick Tour of Linear Algebra and Graph Theory

Quick Tour of Linear Algebra and Graph Theory CS224W: Social and Information Network Analysis Fall 2014 David Hallac Based on Peter Lofgren, Yu Wayne Wu, and Borja Pelato s previous versions Matrices and

### Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε,

2. REVIEW OF LINEAR ALGEBRA 1 Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, where Y n 1 response vector and X n p is the model matrix (or design matrix ) with one row for

### Matrices and Determinants

Chapter1 Matrices and Determinants 11 INTRODUCTION Matrix means an arrangement or array Matrices (plural of matrix) were introduced by Cayley in 1860 A matrix A is rectangular array of m n numbers (or

### 9. The determinant. Notation: Also: A matrix, det(a) = A IR determinant of A. Calculation in the special cases n = 2 and n = 3:

9. The determinant The determinant is a function (with real numbers as values) which is defined for square matrices. It allows to make conclusions about the rank and appears in diverse theorems and formulas.

### 11 a 12 a 13 a 21 a 22 a b 12 b 13 b 21 b 22 b b 11 a 12 + b 12 a 13 + b 13 a 21 + b 21 a 22 + b 22 a 23 + b 23

Chapter 2 (3 3) Matrices The methods used described in the previous chapter for solving sets of linear equations are equally applicable to 3 3 matrices. The algebra becomes more drawn out for larger matrices,

### Department of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in. NUMERICAL ANALYSIS Spring 2015

Department of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in NUMERICAL ANALYSIS Spring 2015 Instructions: Do exactly two problems from Part A AND two

### Evaluating Determinants by Row Reduction

Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.

### 6.4 Determinants and Cramer s Rule

6.4 Determinants and Cramer s Rule Objectives Determinant of a 2 x 2 Matrix Determinant of an 3 x 3 Matrix Determinant of a n x n Matrix Cramer s Rule If a matrix is square (that is, if it has the same

### Lecture 9: Elementary Matrices

Lecture 9: Elementary Matrices Review of Row Reduced Echelon Form Consider the matrix A and the vector b defined as follows: 1 2 1 A b 3 8 5 A common technique to solve linear equations of the form Ax

### 7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Outcomes

The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a 0has a reciprocal b written as a or such that a ba = ab =. Similarly a square matrix A may have an inverse B = A where AB =

### B553 Lecture 5: Matrix Algebra Review

B553 Lecture 5: Matrix Algebra Review Kris Hauser January 19, 2012 We have seen in prior lectures how vectors represent points in R n and gradients of functions. Matrices represent linear transformations

### Linear Algebra: Sample Questions for Exam 2

Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and

### (a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.

1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III

### Determinant: 3.3 Properties of Determinants

Determinant: 3.3 Properties of Determinants Summer 2017 The most incomprehensible thing about the world is that it is comprehensible. - Albert Einstein Goals Learn some basic properties of determinant.

### 5.6. PSEUDOINVERSES 101. A H w.

5.6. PSEUDOINVERSES 0 Corollary 5.6.4. If A is a matrix such that A H A is invertible, then the least-squares solution to Av = w is v = A H A ) A H w. The matrix A H A ) A H is the left inverse of A and

### ENGR-1100 Introduction to Engineering Analysis. Lecture 21

ENGR-1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility

### MATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m

### Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007

Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007 You have 1 hour and 20 minutes. No notes, books, or other references. You are permitted to use Maple during this exam, but you must start with a blank

### ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following

### Calculation in the special cases n = 2 and n = 3:

9. The determinant The determinant is a function (with real numbers as values) which is defined for quadratic matrices. It allows to make conclusions about the rank and appears in diverse theorems and

### Section 5.3 Systems of Linear Equations: Determinants

Section 5. Systems of Linear Equations: Determinants In this section, we will explore another technique for solving systems called Cramer's Rule. Cramer's rule can only be used if the number of equations

### Determinants: summary of main results

Determinants: summary of main results A determinant of an n n matrix is a real number associated with this matrix. Its definition is complex for the general case We start with n = 2 and list important

### Linear Equations and Matrix

1/60 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear

### A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].

Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes