Classify the following system, whether (a) intersecting, (b) parallel, or (c) coinciding lines 1. 3 x 4 y 1 0 3 x 4 y 2 0 3 x 4 y 1 0 6 x 8 y 2 0
Solve the following systems in three variables: 1. 3 x 4 y z 1 2. x y 2 x 4 y 3z 3 3 x 2 y 2 z 0
________
3 y z 1 x 2 z 7
2.
________
3.
2 x 5 y 1 0 5 x 2 y 2 0 2 x y 1 4 x 2 y 3 x 2 y 1 0 2 x y 1
________
4.
________
5.
________
1 x Solve 1 x
2 3 y 3 2 y
Problem solving Form a system of equations from the problems given below. A) (MIXTURE PROBLEM 1) How many pounds of a 35% salt solution and a 14% salt solution should be combined so that a 50 pounds of a 20% solution is obtained? B) (UNIFORM MOTION) Two motorists start at the same time from two places 128 km apart and drive toward each other. One drives 10kph than the other. If they met after 48 minutes (that is, 4/5 hr), find the average speed of each. C) A dietician is preparing a meal consisting of foods A, B, and C as shown in the table below. Fat Protein Carbohydrate If the meal must provide exactly 24 units of fat, 25 Food A 3 2 4 units of protein, and 21 units of carbohydrate, how Food B 2 3 1 many ounces of each food should be used? Food C 3 3 2...
...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...
...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...
...TR 3923 Programming Design in Solving Biology Problems Semester 1, 2011/2012
Elankovan Sundararajan School of Information Technology Faculty of Information Science and Technology
TR 3923 Elankovan Sundararajan 1
Lecture 3
System of LinearEquations
TR 3923
Elankovan Sundararajan
2
Introduction
• Solving sets of linearequations is the most frequently used numerical procedure when realworld situations are modeled. modeled Linearequations are the basis for mathematical models of
1. 2. 2 3. 4. 5. Economics, Computational Biology Comp tational Biolog and Bioinformatics Bioinformatics, Weather prediction, Heat and mass transfer, Statistical analysis, and a myriad of other application.
•
•
TR 3923
The methods for solving ODEs and PDEs also depend on them.
Elankovan Sundararajan
3
System of LinearEquations
• Consider the following general set of n equations in n unknowns: a11 x1 a12 x2 a1n xn c1 : R1
a21 x1 a22 x2 a2 n xn c2 , an 1,1 x1 an 1, 2 x2 an 1,n xn cn 1 , an ,1 x1 an , 2 x2 an ,n xn cn .
Which can be written in matrix form as:
: R2
: R n1 : Rn
A x b.
~ ~
TR 3923 Elankovan Sundararajan 4
where, A is the nxn matrix, ,
a11 a12 a1n a21 a22 a2 n A . a an 2 ann n1 ...
...MODULE  1
LinearEquations
Algebra
5
Notes
LINEAREQUATIONS
You have learnt about basic concept of a variable and a constant. You have also learnt
about algebraic exprssions, polynomials and their zeroes. We come across many situations
such as six added to twice a number is 20. To find the number, we have to assume the
number as x and formulate a relationship through which we can find the number. We shall
see that the formulation of such expression leads to an equation involving variables and
constants. In this lesson, you will study about linearequations in one and two variables.
You will learn how to formulate linearequations in one variable and solve them algebraically.
You will also learn to solve linearequations in two variables using graphical as well as
algebraic methods.
OBJECTIVES
After studying this lesson, you will be able to
•
identify linearequations from a given collection of equations;
•
cite examples of linearequations;
•
write a linearequation in one variable and also give its solution;
•
cite examples and write linearequations in two variables;
•
draw graph of a linearequation in two...
...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...
...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...
...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....
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