LINEAR EQUATION SYSTEMEngineering Mathematics I

LINEAR EQUATION SYSTEM

nnnnnn

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

...

...

...

...

2211

22222121

11212111

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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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