Believe it or not, algebra exists for a reason other than lowering a high school student's grade point average. Systems of linear equations, or a set of equations with two or more variables, are an essential part of finding solutions with only limited information, which happens to be exactly what algebra is. As a required part of any algebra student's life, it is best to understand how they work, not only so an acceptable grade is received, but also so one day the systems can be used to actually find desired information with ease.
There are three main methods of defining a system of linear equations. One way is called a consistent, independent solution. This essentially means that the system has one unique, definite solution. In this situation on a graph, a set of two equations and two variables would be solved as one single point where two lines intersect. It is much the same with three variables and three equations. The only difference is that the point is an intersection of three planes instead of two lines.
Additionally, there are situations where a system of linear equations could be described as consistent, dependent. These systems of linear equations have an infinite number of solutions where a general solution is used to substitute one or two variables for one other selected variable, and solves the other unknown variable or variables in terms of that selected one. Graphically when this system of linear equations is solved for two equations and two variables, the result is lines that coincide, or lay on top of each other, making any point on that line true for the system. A system with three equations and three variables would yield an answer that shows the three planes intersecting on a line or overlaying each other. When the system yields three planes intersecting on a line, all points on the line would make the systems true, and when planes coincide, all points on the coincidental planes would be correct when placed in the system for the variables....
...Solving systems of linearequations
7.1 Introduction
Let a system of linearequations of the following form:
a11 x1
a21 x1
a12 x2
a22 x2
ai1x1 ai 2 x2
am1 x1 am2 x2
a1n xn
a2 n x n
ain xn
amn xn
b1
b2
bi
bm
(7.1)
be considered, where x1 , x2 , ... , xn are the unknowns, elements aik (i = 1, 2, ..., m;
k = 1, 2, ..., n) are the coefficients, bi (i = 1, 2, ..., m) are the free terms of the system. In
matrix notation, this system has the form:
Ax b ,
(7.2)
where A is the matrix of coefficients of the system (the main matrix), A = [aik]mn, b is the
column vector of the free terms, bT [b1 , b2 , ... , bm ] , x is the column vector of the
unknowns, xT [ x1 , x2 , ... , xn ] ; the symbol () T denotes transposition.
It is assumed that aik and bi are known numbers. An ordered set, {x1, x2, ..., xn}, of real
numbers satisfying (7.1) is referred to as the solution of the system, and the individual
numbers, x1, x2, ..., xn, are roots of the system.
A system of linearequations is:
consistent  if it has at least one solution. At the same time it can be

determined  if it has exactly one, unique solution,
undetermined  if it has...
...Semester MATH 1111 Linear Algebra Chapter 1: Matrices and Systems of Equations
Coverage of Chapter 1: Skip Application 3 in Section 1.4. Skip ‘Triangular Factorisation’ in Section 1.5.
A.
Solving Equations
1.
We are all familiar with solving equations. Illustrate how the following equations can be solved, and then raise and answer some theoretical and/or practical questions concerning the process of solution. (a)
3x 1 8
2 x x 1 (b) 1 2 3 x1 x2 4
(c) (d)
x 4 5 x 2 36 0
x2 x4
B.
Systems of LinearEquations
(Ref: Sections 1.1 and 1.2)
2.
Your first experience of solving systems of equations was probably to deal with a system of two equations in two unknowns. Can such equations always be solved? Are there any special cases?
1
3.
The system mentioned in the previous question is usually called a 2 2 system. How about n n systems in general? (Note: An m n system is a system consisting of m __________ and n __________.)
4.
Solve the following systems of equations. (a)
x1 2 x2 x3 3 x2 x3 2 2 x3 8 x1 2 x2 x3 3 (b) 3 x1 x2 3 x3 1 2 x1 3 x2 x3 4
5....
...The purpose of this project is to solve the game of Light’s Out! by using basic knowledge of Linear algebra including matrix addition, vector spaces, linear combinations, and row reducing to reduced echelon form. 
Lights Out! is an electronic game that was released by Tiger Toys in 1995. It is also now a flash game online. The game consists of a 5x5 grid of lights. When the game stats a set of lights are switched to on randomly or in a pattern. Pressing one light will toggle it and the lights adjacent to it on and off. The goal of the game is to switch all the lights off in as few button presses as possible. In the folling examples, 1 will represent a “on” light and 0 will represent an “off” light. Yellow represents a button pressed and changed and green represents a button that was not pressed but was changed as a result of the pressed button.
Example 1
1  1  0  1  1 
1  0  1  0  1 
0  1  1  1  0 
1  0  1  0  1 
1  1  0  1  1 
0  0  0  1  1 
0  0  1  0  1 
0  1  1  1  0 
1  0  1  0  1 
1  1  0  1  1 
0  0  0  1  1 
0  0  0  0  1 
0  0  0  0  0 
1  0  0  0  1 
1  1  0  1  1 
Starting Grid Pressing button 1 Pressing button 18
Example 2
1  1  0  1  1 
1  0  1  0  1 
0  1  1  1  0 
1  0  1  0  1 
1  1  0  1  1 
0  0  0  1  1 
0  0  1  0  1 ...
...simultaneous equations using matrix method.
3x + y = 4
4x + 3y = 7(5 marks)
Find the value of K which makes a singular matrix.(3 marks)
3 1
4 2
4 K 0
Calculate the cross product of the vector U = 2i – 3j – k and V = i + 4j – 2k.(3 marks)
Given the matrices.
2 5 3 2 0
A = 3 1 and B = 1 1 4
4 2 5 5 5
Compute:
ATB(3 marks)
tr (AB)(1 mark)
(e) Determine if (2, 1) is in the set generated by = (3, 1), (2, 2) (5 marks)
Question Two (20 marks)
Let T: R2 R2 be defined by T(x, y) = (x + y, x). Show that T is a linear transformation.(7 marks)
Find the basis and dimension of the row space of the matrix.(6 marks)
2 1 3
A= 1 1 5
1 2 2
Compute A1 using row reduction method.(7 marks)
1 4 3
A= 1 2 0
2 2 3
Question Three (20 marks)
Find x, y and z by use of determinants.(10 marks)
X – 3y – 2z – 6=0
2x – 4y – 3z – 8 =0
3x + 6y + 8z + 5 =0
Determine S= (1, 0), (0, 1) if linearly independent or dependent.
(3 marks)
Show that S= (1, 0, 0), (0, 1, 0), (0, 0, 1) is a basis for R3 (7 marks)
Question Four (20 marks)
Consider the matrix
1 2 3 1 2
A= 2 4 4 6 10
3 6 6 9 13
Reduce to an echelon form.
Reduce to its row canonical from.
State the rank of the matrix(10 marks)
Using...
...Universitatea Politehnica Bucuresti  FILS
Systems of Differential Equations and Models in Physics, Engineering and Economics
Coordinating professor: Valeriu Prepelita
Bucharest,
July, 2010
Table of Contents
1. Importance and uses of differential equations 4
1.1. Creating useful models using differential equations 4
1.2. Reallife uses of differential equations 5
2. Introduction to differentialequations 6
2.1. First order equations 6
2.1.1. Homogeneous equations 6
2.1.2. Exact equations 8
2.2. Second order linearequations 10
3. Systems of differential equations 14
3.1. Systems of linear differential equations 16
3.1.1. Systems of linear differential equations with constant coefficients 22
3.2. Systems of first order equations 27
3.2.1. General remarks on systems 27
3.2.2. Linearsystems. Case n=2 31
3.2.3. Nonlinear systems. Volterra’s Preypredator equations 38
3.3. Critical points and stability for linearsystems 44
3.3.1. Bounded input bounded output stability 44
3.3.2. Critical points 44
3.3.3. Methods of...
...LINEAR ALGEBRA
Paul Dawkins
Linear Algebra
Table of Contents
Preface............................................................................................................................................. ii Outline............................................................................................................................................ iii Systems of Equations and Matrices.............................................................................................. 1
Introduction ................................................................................................................................................ 1 Systems of Equations ................................................................................................................................. 3 Solving Systems of Equations .................................................................................................................. 15 Matrices .................................................................................................................................................... 27 Matrix Arithmetic & Operations .............................................................................................................. 33 Properties of Matrix Arithmetic and the Transpose...
...most distinct and beautiful statement of any truth must atlast take the Mathematical form’ Thoreau. Among the Nobel Laureates in Economics more than 60% were Economists who have done pioneering work in Mathematical Economics.These Economists not only learnt Higher Mathematics with perfection but also applied it successfully in their higher pursuits of both Macroeconomics and Econometrics. A Mathematical formula (involving stochastic differential equations) was discovered in 1970 by Stanford University Professor of Finance Dr.Scholes and Economist Dr.Merton.This achievement led to their winning Nobel Prize for Economics in 1997.This formula takes four input variablesduration of the option,prices,interest rates and market volatilityand produces a price that should be charged for the option.Not only did the formula work ,it transformed American Stock Market. Economics was considered as a deductive science using verbal logic grounded on a few basic axioms.But today the transformation of Economics is complete.Extensive use of graphs,equations and Statistics replaced the verbal deductive method.Mathematics is used in Economics by beginning wth a few variables,gradually introducing other variables and then deriving the inter relations and the internal logic of an economic model.Thus Economic knowledge can be discovered and extended by means of mathematical formulations. Modern Risk Management including Insurance,Stock Trading and Investment depend...
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