LINEAR EQUATION SYSTEM Engineering Mathematics I
LINEAR EQUATION SYSTEM
nnnnnn
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
...
...
...
...
2211
22222121
11212111
bAx Engin
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nnnnnn
n
n
n
b
b
b
b
aaaa
aaaa
aaaa
aaaa
...
...
...............
...
...
...
3
2
1
321
3333231
2232221
1131211
2
Augmented matrix A
GAUSS ELIMINATION (1)
nnnnnn
n
n
n
b
b
b
b
aaaa
aaaa
aaaa
aaaa
...
...
...............
...
...
...
3
2
1
321
3333231
2232221
1131211
Eliminate
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nnn
n
n
n
b
b
b
b
a
aa
aaa
aaaa
...
...000
...............
...00
...0
...
3
2
1
333
22322
1131211
3
Upper triangular matrix
GAUSS ELIMINATION (2)
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1,1
,1111,111,1
nn
nnnnnnnnnnnn
nn
nnnnnn
a
xabxbxaxa
a
bxbxa
kk
n
kjjkjk
k a
xab
x
1
Backward substitution
44
EXAMPLE 1
132
3344
532
321
321
321
xxx
xxx
xxx
6260
7120
5132
1132
3344
5132
Pivot element
* Replace 2nd eq.
(2nd eq.) – 2x(1st eq.)
* Replace 3rd eq.
(3rd eq.) + 1x(1st eq.)
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EXAMPLE 1
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15500
7120
2121
6260
7120
5132
* Replace 3rd eq.
(3rd eq.) + 3x(2nd eq.)
Upper triangle
3
2
5
3
2
1
x
x
x
POSSIBILITIES (1)
Linear equation system has three possibilities of solutions
Many solutions No solution Unique solution
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1;0;1
3300
1110
0111
1213
1132
0111
321
xxx
0000
0000
6330
4211
6321
2112
4211
321
xxx
1000
1000
6330
4211
7321
2112
4211
321
xxx
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EXAMPLE 2 Kirchhoff's current Law (KCL):
At any point of a circuit, the sum of the inflowing currents equals the sum of out flowing currents.
Kirchhoff's voltage law (KVL): In any closed loop, the sum of all voltage drops equals the impressed electromotive force.
P
Q
80V 90V
20 Ohms 10 Ohms
15 Ohms
i1
i2
i3
10
Oh
ms
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EXAMPLE #2
P
Q
80V 90V
20 Ohms 10 Ohms
15 Ohms
i1
i2
i3
10 O
hms
8001020
9025100
0111
0111
Node P: i1 – i2 + i3 = 0 Node Q: -i1 + i2 –i3 = 0 Right loop: 10i2 + 25i3 = 90 Left loop: 20i1 + 10i2 = 80
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LINEAR INDEPENDENCE Let a1, …, am be any vectors in a vector
space V. Then an expression of the form c1a1 + … + cmam (c1, …, cm any scalars)is called linear combination of these vectors.
The set S of all these linear combinations is called the span of a1, …, am.
Consider the equation: c1a1 + … + cmam = 0 If the only set of scalars that satisfies the
equation is c1 = … = cm = 0, then the set of vectors a1, …, am are linearly independent.
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LINEAR DEPENDENCE Otherwise, if the equation also holds with
scalars c1, …, cm not all zero (at least one of them is not zero), we call this set of vectors linearly dependent.
Linear dependent at least one of the vectors can be expressed as a linear combination of the others.
If c1 ≠ 0,a1 = l2a2 + … + lmam where lj = -cj/c1
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EXAMPLE 3 Consider the vectors:
i = [1, 0, 0], j = [0, 1, 0] and k = [0, 0, 1], and the equation: c1i + c2j + c3k = 0
Then: [(c1i1+c2j1+c3k1), (c1i2+c2j2+c3k2), (c1i3+c2j3+c3k3)] = 0
[c1i1, c2j2, c3k3] = 0
c1 = c2 = c3 = 0
Consider vectors a = [1, 2, 1], b = [0, 0, 3], d = [2, 4, 0]. Are they linearly independent?
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RANK OF A MATRIX There are some possibilities of solutions of
linear equation system: no solution, single solution, many solution.
Rank of matrix a tool to observe the problems of existence and uniqueness.
The maximum number of linearly independent row vectors of a matrix A is called the rank of A.
Rank A = 0, if and only if A = 0.
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EXAMPLE 4
1502121
5424426
2203
A
Matrix A above has rank A = 2
Since the last row is a linear combination of the two others (six times the first row minus ½ times the second), which are linearly independent.
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EXAMPLE 5
a
b
cd
b b
Linearly dependent Linearly independent
Rank = 1 Rank = 2
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EXAMPLE 6
a b
c
a
-b
c
Linearly dependentRank = 2
c = ka + sb a = (1/k)c - (s/k)b
Linearly dependentRank = 2
b-a
c
b = (1/s)c - (k/s)a
Linearly dependentRank = 2
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EXAMPLE 7
i
j
k
Linearly independentRank = 3
d e
f
d = pe
Linearly dependentRank = 1
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SOME NOTES
For a single vector a, then the equation ca = 0, is satisfied if:c = 0, and a ≠ 0 a is linearly
independenta = 0, there will be some values c ≠ 0 a
is linearly dependent.
Rank A = 0, if and only if A = 0.Rank A = 0 maximum number of linearly
independent vectors is 0. If A = 0, there will be some values c1, …,
cm which are not equal to 0, then vectors in A are linearly dependent.
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RANK OF A MATRIX (2) The rank of a matrix A equals the maximum
number of linearly independent column vectors of A.
Hence A and AT have the same rank.
If a vector space V is such that it contains a linearly independent set B of n vectors, whereas any set of n + 1 or more vectors in V is linearly dependent, then V has n dimension and B is called a basis of V.
Previous example: vectors i, j, and k in vector space R3. R3 has 3 dimension and i, j, k is the basis of R3.
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GENERAL PROPERTIES OF SOLUTIONS
A system of m linear equations has solutions if and only if the coefficient matrix A and the augmented matrix Ã, have the same rank.
If this rank r equals n, the system has one solution.
If r < n, the system has infinitely many solutions.
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