Date post: | 04-Apr-2018 |
Category: |
Documents |
Upload: | ahmed-alhashmi |
View: | 220 times |
Download: | 0 times |
of 34
7/29/2019 math204_la_Topic-1 (1).ppt
1/34
SYSTEMS OF LINEAR
EQUATIONS
Topic-1
7/29/2019 math204_la_Topic-1 (1).ppt
2/34
Real-Life Examples
Example 1: Very simple case
I am thinking of a number. If I add 3 to that number, I will get 7.What is the number?
Let x be the number in my mind
Adding 3 to x gives 7 The equation is 3 + x = 7
Example 2:
Taxi drivers usually charge a an initial fixed fee as part of usingtheir services. Then, for each mileage, they charge a certain
amount
Say for instance, the initial fee is 10 AED and each mileage cost 2AED
How will be model the total cost as an equation?
7/29/2019 math204_la_Topic-1 (1).ppt
3/34
Real-Life Examples (cont.)
Example 2 (cont.)
Solution
Let y be the total cost
Let N be the number of miles
So, the total cost = initial cost + number of miles x chargesper mile
That is, y = 4 + N x 2
So what can we infer?
These equations model the relationship betweentwo variables and the effect that a change on onevariable has on the other
7/29/2019 math204_la_Topic-1 (1).ppt
4/34
Why is this important?
Think of the following situations
the relationship between the price of a product and the quantityconsumers are willing to buy demand planning
how much of their products to sell at a price that maximizes profits supply curve
how does the change in one currency value affect the otherforeign exchange
In the more complex case of a manufacturers profit depends on:
Material cost, labor costs, transportation, overhead etc.
A realistic modeling of these relationships would involve all thesevariables
Mathematically, profit now is a function of several variables
http://www.ehow.com/facts_6027891_examples-equations-used-real-life.html
7/29/2019 math204_la_Topic-1 (1).ppt
5/34
Linear Algebra
In linear algebra we study the simplest functions of several variables, the onesthat are linear
Linear equations:
In n unknowns, linear equation is an equation that is of the form:
Which is the unknown?
x1, x
2, x
n
Then, what are a1
, a2
, an
and b?
Known, a1, a
2, a
n coefficients and b constant
The solution of a linear equation is any sequence s1, s2, sn of number suchthat the substitution of x
1= s
1, x
2= s
2, x
n= s
nsatisfies the equation
7/29/2019 math204_la_Topic-1 (1).ppt
6/34
Why linear?
Linear because:
the unknowns only appear to the first power
there arent any unknowns in the denominator
of a fraction. there are no products and/or quotients of
unknowns.
In the other way:
Unknowns only occur in numerators, they areonly to the first power and there are no productsor quotients of unknowns.
7/29/2019 math204_la_Topic-1 (1).ppt
7/34
Linear System
Systems of linear equations
collection of two or more linear equationsinvolving the same unknowns
Consider,
7/29/2019 math204_la_Topic-1 (1).ppt
8/34
Linear System
Unknowns: x1, x
2, x
n; coefficients: a
11, a
12,
Constants: b1, b
2, b
m
Here, the solution set to a system with nunknowns is a set of numbers so that thesubstitution of x
1= t
1, x
2= t
2, x
n= t
n
satisfies all the equation
7/29/2019 math204_la_Topic-1 (1).ppt
9/34
Graphical Interpretation
Let us consider:
Subject to
Coefficients of each equation not both zeros
Three possible solutions:
222121
1212111
bxaxa
bxaxa
7/29/2019 math204_la_Topic-1 (1).ppt
10/34
Geometrical Interpretation
Now lets see:
Represents planes
Possible solutions include:
The two planes might be coincident or they canintersect in a line;
Infinite solutions
The two planes can be parallel No solution
How about the same case for (3 x 3) system?
323222121
1313212111
xaxaxa
xaxaxa
7/29/2019 math204_la_Topic-1 (1).ppt
11/34
Geometric Interpretation
In general an (m x n) system of linearequations may have (a) infinitely manysolutions, (b) no solution or (c) a uniquesolution
7/29/2019 math204_la_Topic-1 (1).ppt
12/34
Classification of Linear Systems
There is a unique solution, in which case we say that thesystem is consistent and nonsingular. For a nonsingularsystem we must have that m = n, although the converse isnot true.
There are no solutions, in which case we say that thesystem is singularand inconsistent. Typically (but notalways!) an inconsistent system is one which is over-determined that is, where m > n.
There are infinitely many solutions, in which case we say
that the system is consistent but singular. Typically (but notalways!) a consistent, singular system is one which isunder-determined that is, where m < n.
7/29/2019 math204_la_Topic-1 (1).ppt
13/34
Differentiation
Consistent system has atleast one solution
Inconsistent system has no solution
A consistent system has either one solution or an infinitenumber of solutions (i.e. it is not possible for a linear
system to have, for example, exactly five solutions)
Homogeneous all constant terms are zero; otherwise non-homogeneous
Just-determined system: same equations and unknowns
Over-determined system: more equations than unknowns
Under-determined system: fewer equations than unknowns
7/29/2019 math204_la_Topic-1 (1).ppt
14/34
Matrices an introduction
Matrices provide a natural symbolic language to describelinear systems
Appropriate, convenient and powerful framework foranalyzing and solving linear problems
In general, an (m x n) matrix is a rectangular array of theform:
m rows; n columns
Have we seen the above notation somewhere?
mnmm
n
n
aaa
aaa
aaa
A
...
....
...
...
21
22221
11211
7/29/2019 math204_la_Topic-1 (1).ppt
15/34
Linear System - Matrixrepresentation
Let re-think the general equation of a linear system.
How about?
B is called the augmented matrix for the system denoted as[A | b]
mmnmm
n
n
baaa
baaa
baaa
B
...
.....
.....
...
...
21
222221
111211
7/29/2019 math204_la_Topic-1 (1).ppt
16/34
Linear System MatrixRepresentation
Where, A is called the coefficient matrix ofthe form,
And b is,
mnmm
n
n
aaa
aaa
aaa
A
...
....
...
...
21
22221
11211
mb
b
b
b.
2
1
7/29/2019 math204_la_Topic-1 (1).ppt
17/34
Example
Let us consider,
Here,
23
12
22
321
321
321
xxx
xxx
xxx
213
112
121
A
5
1
2
b
2131112
2121
]|[ bAB
7/29/2019 math204_la_Topic-1 (1).ppt
18/34
Solving Linear System ofEquations
Two steps involved in solving an (m x n) system ofequations are:
Reduction of the system (that is, elimination of variables)
Goal of reduction is to yield equivalent system of equations
Description of the set of solutions
What are we trying to achieve?
Obtain Equivalence
Two systems of linear equations in n unknowns areequivalent provided they have the same set of solutions
How to perform reduction? Elementaryoperations...
7/29/2019 math204_la_Topic-1 (1).ppt
19/34
Elementary Operations
High school method of solving linear equations
Possible elementary operations
Interchange two equations
Multiply an equation by non-zero scalarAdd a constant multiple of one equation to another
That is,
EiEj interchange i and j equations
kEi multiply ith equation by non zero scalar k
Ei+kEj k times the jth equation added to the ith equation
7/29/2019 math204_la_Topic-1 (1).ppt
20/34
Example
Let us consider,
Now the elementary operation E2 + E1 willproduce,
Now (1/3)E2 will give us,
Finally, E1 E2 will produce,
2
5
21
21
xx
xx
93
5
2
21
x
xx
3
5
2
21
x
xx
3
2
2
1
x
x
7/29/2019 math204_la_Topic-1 (1).ppt
21/34
Row Operations
But we are in a university, so lets think matrices
Elementary operations on rows of the matrix?
So, elementary row operations are:
Interchange 2 rows
Multiply a row by a non-zero scalar
Add a constant multiple of one row to another
That is,RiRj interchange i and j rows
kRi multiply ith row by non zero scalar k
Ri+kRj k times the jth row added to the ith row
7/29/2019 math204_la_Topic-1 (1).ppt
22/34
Solving a System of LinearEquations
Thus we can solve a linear system with thefollowing steps
Form the augmented matrix B for the system
Use elementary row operations to transform B to rowequivalent matrix Cthis represents a simpler system
Solve the simpler system represented by C
Gauss-Jordan Elimination method reductionprocess
Givensystem ofequations
Build theAugmented
Matrix
ReducedMatrix
ReducedSystem ofEquations
Solution
7/29/2019 math204_la_Topic-1 (1).ppt
23/34
Note:
Why use matrix?
Because we do not need the list of variables, the sequence ofsteps in the matrix method is easier to perform and record.
The objective of the Gauss-Jordan reductionprocess is to obtain a system of equations to apoint where we can immediately describe thesolution.
How do we know when the system has beensimplified as much as it can be?
The system has been simplified as much as possible when itis in the row echelon form
7/29/2019 math204_la_Topic-1 (1).ppt
24/34
Echelon Form
An (m x n) matrix B is in echelon form if:
All rows that consist entirely of zeros are grouped together atthe bottom of the matrix
In every non-zero row, the first entry (counting left to right) is
a 1 (call this a leading 1)
If the (i+1)st row contains non-zero entries, then the first non-zero entry is in a column to the right of the first non-zero entryin the ith row.
A matrix that is in echelon form is in reducedechelon form provided that the first non-zero entryin any row is the only non-zero entry in its column
7/29/2019 math204_la_Topic-1 (1).ppt
25/34
Examples
Echelon form
Reduced Echelon form
0000000
3100000
2121000
2348100
0203411
A
10000
56100
34110
B
3100
1010
2001
A
00000
43100
11021
B
7/29/2019 math204_la_Topic-1 (1).ppt
26/34
Reduction to Echelon Form
Conversion process to the Reduced Echelon form for an (mx n) matrix (cont.):
Locate the first (left most) column that contains a nonzero entry
If necessary, interchange the first row with another so that the firstnonzero column has a nonzero entry in the first row.
If a denotes the leading nonzero entry in row one, multiply each entry inrow one by (1/a). Thus leading nonzero entry in row one is 1)
Add appropriate multiples of row one to each of the remaining rows sothat every entry below the leading 1 in row one is zero.
Once done, ignore row one temporarily, and repeat above process ofthe submatrix this will produce the echelon form
To get to the reduced echelon form, proceed upward, add multiples ofeach nonzero row to the rows above in order to zero all entries abovethe leading 1
7/29/2019 math204_la_Topic-1 (1).ppt
27/34
Example
1101000
4820000
91131000
11831410
1000000
0410000
0101000
0300410
211761820
3324931230
91131000
4820000
211761820
4820000
91131000
3324931230
211761820
4820000
91131000
11831410
81230000
4820000
91131000
2300410
81230000
2410000
91131000
2300410
2000000
2410000
3101000
2300410
1000000
2410000
3101000
2300410
R1R3
R1(1/3)R1
R4->R4+2R1
R1->R1+R2R4->R4+R2
R3->(1/2)R3
R2->R2+(-3)R3R4->R4+(-3)R3
R4->(1/2)R4
R1->R1+2R4R2->R2+(-3)R4R3->R3+(-2)R4
7/29/2019 math204_la_Topic-1 (1).ppt
28/34
Reduction to Echelon Form
Let B be an (m x n) matrix. There is a unique (m x n) matrixsuch that:
C is in reduced echelon form
C is row equivalent to B
Solving System of Linear Equations
Create the augmented matrix of the system
Transform the augmented matrix to reduced echelon
Decode the reduced matrix to obtain equivalent system ofequations
By examining the reduced system, describe the solution ofthe original system
7/29/2019 math204_la_Topic-1 (1).ppt
29/34
Solve
612881063
1063
13522
28114342
54321
543
54321
54321
xxxxx
xxx
xxxxx
xxxxx
Recognizing an Inconsistent
7/29/2019 math204_la_Topic-1 (1).ppt
30/34
Recognizing an InconsistentSystem
Let [A|b] be the augmented matrix for an (m x n)linear system of equations.
If [A|b] is in reduced echelon form and if the lastnonzero row of [A|b] has its leading 1 in the last
column, then system of equations has no solution.
i.e.
last nonzero row of [A|b] has the form:
[0 0 0 0 1]
The equation for the row is 0x1 + 0x2+ + 0xn = 1
Consistent Systems of Linear
7/29/2019 math204_la_Topic-1 (1).ppt
31/34
Consistent Systems of LinearEquations
Goal is to deduce as much information as possible about the solutionset without actually solving the system.
Let [A|b] be the augmented matrix representing a (m x n) linear system
[A|b] can be simplified to the reduced row echelon form to a rowequivalent matrix [C|d]
[C|d], we can say:
Is inconsistent if and only if [C|d] has a row of the form [0 0 0 0 0 1]
Every variable corresponding to a leading 1 in [C|d] is a dependent variable (i.e.leading 1 variables can be expressed in terms of the independent or nonleading 1variables
If r denote the number of nonzero rows in [C|d],
then r
7/29/2019 math204_la_Topic-1 (1).ppt
32/34
Homogeneous Systems
Homogeneous system has
the form:
Such a linear system is
always consistent.i.e. x1=x2=x3=xn=0 is a solution
This solution is called the trivial or zero solution
Any other solution to such a homogeneous system is called
nontrivial solution
A homogeneous system can have a unique trivial solutionor also has non-trivial (infinitely many) solutions.
7/29/2019 math204_la_Topic-1 (1).ppt
33/34
Example
What are the possibilities for the solution setof
02663
0342
032
4321
4321
4321
xxxx
xxxx
xxxx
7/29/2019 math204_la_Topic-1 (1).ppt
34/34
Solution
02663
01342
03121
07300
05100
03121
08000
05100
08021
01000
00100
00021
0
0
02
4
3
21
x
x
xx
0
0
2
4
3
21
x
x
xx
R2-2R1, R3-3R1 R3-3R2,R1-R2 (1/8)R3,R1-8R3,R2+5R3