2.1 Introduction to Linear Systems

Save this PDF as:

Size: px
Start display at page:

Transcription

1 2.1 Introduction to Linear Systems Introduction to Linear Systems A line in the y-plane can be represented by an equation of the form : a 1 + a 2 y = b. This equation is said to be linear in the variables and y. For eample, + 3y = 6. (Note if = 0 then 3y = 6 so. Likewise y = 0 when = 6. Thus the line passes through the points (0, 2) and (6, 0).) In general we have the definition: Definition: A linear equation in n variables, 1, 2,..., n, has the form a a a n n = b, where a 1, a 2,..., a n, b are real scalars (i.e. constants). The variables 1, 2,..., n can also be called the unknowns. Eample: + 2y + 3z = 4 (or = 4) is a linear equation representing a plane. Points in the plane can be found by taking particular values for two of the variables and using the equation to solve for the third. For instance, if = 0 and y = 0, then 3z = 4 or z = 4/3; similarly, if = 0 and z = 0, then ; finally, if y = 0 and z = 0, then = 4. Therefore, + 2y + 3z = 4 is the equation of the plane (in 3-d space) which passes through the points (0, 0, 4/3), (0, 2, 0) and (4, 0, 0). Remark: A linear equation does not involve powers of variables or products of variables. Nor do the variables appear as arguments of trigonometric, eponential or logarithmic functions. Eamples: = 3 + 1, + 5y = 2 and = 0 are linear equations. Whereas = 4, sin 2 = 0 and = 1 are not. Definiton: A solution of a linear equation is a sequence of n numbers s 1, s 2,..., s n such that when 1 = s 1, 2 = s 2,..., n = s n is substituted into the equation, the equation is satisfied. The set of all solutions is called the solution set or the general solution. Eample: Find the solution set of + 3y = 6. Solution: To find solutions, an arbitrary value must be assigned to one of the variables. First assign an arbitrary value t. Then = t and t + 3y = 6. Thus 3y = 6 t or t/3. The solution set is given by = t, t/3; t R. Then giving t a particular value yields a particular solution of the equation: e.g. t = 0; = 0, 0 = 2 or (, y) = (0, 2). We could also have assigned y an arbitrary parameter: y = s, = 6 3s; s R. In this case, giving s the particular value s = 2 yields the same solution as before: y = s = 2, = 6 3(2) = 0 or (, y) = (0, 2). Eample: Find the solution set of + 2y + 3z = 4. Solution: To find solutions in this case, arbitrary values must be assigned to any two of the variables and then the equation is solved for the third variable. For instance, if = t, y = s, then t + 2s + 3z = 4 giving 3z = 4 t 2s and z = 4/3 t/3 2s/3 (s, t R). Then each pair of values s, t gives a point in the plane, e.g. t = 2, s = 1 gives = 2, y = 1, z = 4/3 2/3 2/3 = 0.

2 2.1 Introduction to Linear Systems 2 Definition: A system of linear equations or linear system is a finite set of linear equations in the variables 1, 2,..., n. A solution of a linear system is a sequence of n numbers s 1, s 2,..., s n if 1 = s 1, 2 = s 2,..., n = s n is a solution of every equation in the system. Eample: + y = 4 2 y = 5 This is a system of 2 equations in 2 unknowns (called a 2 2 linear system). = 3, y = 1 is a solution of the system since these values satisfy both equations: = 4 and 2(3) 1 = 5. = 2, is not as it is a solution of the first equation only: = 4 but 2(2) 2 = 2 5. Remark: Not all systems of equations have solutions. For instance, consider the system: Multiply the second equation by 1/2: + y = y = 6 + y = 4 + y = 3 These are contradictory equations and thus there is no solution to the system. Definition: A system of equations that has no solution is said to be inconsistent. If there is at least one solution it is called consistent. Remark: There are only 3 possibilities: every system of linear equations has either no solutions, eactly one solution or infinitely many solutions. To see this consider, for instance, a system of 2 equations in the variables and y (a 2 2 system). Each of the equations represents a line and there are only three possible cases: (i) y l 1 l 2

3 2.1 Introduction to Linear Systems 3 (ii) y l 2 l 1 (iii) y l 1 l 2 (i) The lines are parallel, (ii) The lines intersect at a single point, (iii) The lines coincide. An arbitrary system of m equations in n variables (an m n linear system) can be written as a a a 1n n = b 1 a a a 2n n = b =. a m1 1 + a m a mn n = b m The double subscript shows where the coefficient appears in the system: a ij is the coefficient in the ith equation multiplying variable j. In order to determine the solutions of a linear system, it is usually most convenient to write the system in matri form. A matri is a rectangular array of numbers and the augmented matri for a linear system is given by Eamples: System System 2 + 3y = 7 4 y = 2 + y 5z = 14 3y + 2z = 1 4y + z = 6 a 11 a a 1n b 1 a 21 a a 2n b a m1 a m2... a mn b m Augmented matri Augmented matri ( )

4 2.1 Introduction to Linear Systems 4 The basic method behind solving a system of equations is to replace the given system by a new system which has the same solution set but is easier to solve. The new system is obtained by applying a sequence of operations to the original system in order to eliminate the unknowns. The three types of operations allowed are 1. Multiplying an equation by a non-zero constant. 2. Swapping two equations. 3. Adding a multiple of one equation to another equation. As the rows of the augmented matri for a system correspond to the equations in the system, these operations correspond to the following operations performed on the matri: 1. Multiplying a row by a non-zero constant. 2. Swapping two rows. 3. Adding a multiple of one row to another row. These are called elementary row operations. Eample: + 3y + 6z = 13 2y + 3z = E 2 E 2 E 1 R 2 R 2 R 1 2y + 3z = E 3 E 3 2E 2 R 3 R 3 2R 2 5z = E E 3 R R 3 A value for one of the variables, z, has been found and thus the solution of the system can now be obtained by using this to find the values of the other variables: substituting in E 2 gives y 12 = 10 or, then substituting, in E 1 gives = 3 or = 1. Alternatively, we could carry on in a similar manner to before to obtain the solution:

5 2.1 Introduction to Linear Systems 5 E 2 E 2 4E 3 R 2 R 2 4R 3 E 1 E 1 2E 3 R 1 R 1 2R 3 + 2y = E 1 E 1 2E 2 R 1 R 1 2R 2 = Solution: = 1,,.

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

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

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

Linear Equations in Linear Algebra

1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A

Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.

SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

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

SYSTEMS OF LINEAR EQUATIONS

SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations

2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are

Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented

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 +

Linear Equations ! 25 30 35\$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development

MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!

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

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

Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes

Jim Lambers MAT 169 Fall Semester 009-10 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that

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

Lecture 1: Systems of Linear Equations

MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

Row Echelon Form and Reduced Row Echelon Form

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

5.5. Solving linear systems by the elimination method

55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve

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

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient

Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear

Solving Systems of Linear Equations; Row Reduction

Harvey Mudd College Math Tutorial: Solving Systems of Linear Equations; Row Reduction Systems of linear equations arise in all sorts of applications in many different fields of study The method reviewed

Math 215 HW #1 Solutions

Math 25 HW # Solutions. Problem.2.3. Describe the intersection of the three planes u+v+w+z = 6 and u+w+z = 4 and u + w = 2 (all in four-dimensional space). Is it a line or a point or an empty set? What

The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

1.5 SOLUTION SETS OF LINEAR SYSTEMS

1-2 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as

Systems of Linear Equations

Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11-]. Main points in this section: 1. Definition of Linear

Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

5 Systems of Equations

Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

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

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

10.1 Systems of Linear Equations: Substitution and Elimination

726 CHAPTER 10 Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Sustitution and Elimination PREPARING FOR THIS SECTION Before getting started, review the following: Linear Equations

Linearly Independent Sets and Linearly Dependent Sets

These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

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

Lecture Notes 2: Matrices as Systems of Linear Equations

2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably

Matrices, Determinants and Linear Systems

September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we

8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes

Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us

4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

Solution of Linear Systems

Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start

Systems of Linear Equations and Inequalities

Systems of Linear Equations and Inequalities Recall that every linear equation in two variables can be identified with a line. When we group two such equations together, we know from geometry what can

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

Solving simultaneous equations using the inverse matrix

Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix

Lines and Planes in R 3

.3 Lines and Planes in R 3 P. Daniger Lines in R 3 We wish to represent lines in R 3. Note that a line may be described in two different ways: By specifying two points on the line. By specifying one point

4.3-4.4 Systems of Equations

4.3-4.4 Systems of Equations A linear equation in 2 variables is an equation of the form ax + by = c. A linear equation in 3 variables is an equation of the form ax + by + cz = d. To solve a system of

Projective Geometry. Projective Geometry

Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,

3. Solve the equation containing only one variable for that variable.

Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated

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

Arithmetic and Algebra of Matrices

Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational

Math 2443, Section 16.3

Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.

DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms

4.6 Linear Programming duality

4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

Reasoning with Equations and Inequalities

Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.

Texas Instruments TI-83, TI-83 Plus Graphics Calculator I.1 Systems of Linear Equations

Part I: Texas Instruments TI-83, TI-83 Plus Graphics Calculator I.1 Systems of Linear Equations I.1.1 Basics: Press the ON key to begin using your TI-83 calculator. If you need to adjust the display contrast,

MAT 1341: REVIEW II SANGHOON BAEK

MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and

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

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

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

Section 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)

Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

Notes from February 11

Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The

Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

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

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

Introduction to Diophantine Equations

Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field

Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

LINEAR EQUATIONS IN TWO VARIABLES

66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

Sets and Counting. Let A and B be two sets. It is easy to see (from the diagram above) that A B = A + B A B

Sets and Counting Let us remind that the integer part of a number is the greatest integer that is less or equal to. It is denoted by []. Eample [3.1] = 3, [.76] = but [ 3.1] = 4 and [.76] = 6 A B Let A

SOLVING LINEAR SYSTEMS

SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis

Integrating algebraic fractions

Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D Welcome everybody. We continue the discussion on 2D

Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col

Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

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

Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

Taylor Series and Asymptotic Expansions

Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.

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

TWO-DIMENSIONAL TRANSFORMATION

CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

Lecture 14: Section 3.3

Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

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

Systems of Linear Equations in Three Variables

5.3 Systems of Linear Equations in Three Variables 5.3 OBJECTIVES 1. Find ordered triples associated with three equations 2. Solve a system by the addition method 3. Interpret a solution graphically 4.

The Matrix Elements of a 3 3 Orthogonal Matrix Revisited

Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, Three-Dimensional Proper and Improper Rotation Matrices, I provided a derivation