Linear Equation System

LINEAR EQUATION SYSTEMEngineering Mathematics I

LINEAR EQUATION SYSTEM

nnnnnn

nn

nn

bxaxaxa

bxaxaxabxaxaxa

......

......

2211

22222121

11212111

bAx Engineering Mathem

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nnnnnn

n

n

n

b

bbb

aaaa

aaaaaaaaaaaa

......

........................

3

2

1

321

3333231

2232221

1131211

2

Augmented matrix A

GAUSS ELIMINATION (1)

nnnnnn

n

n

n

b

bbb

aaaa

aaaaaaaaaaaa

......

........................

3

2

1

321

3333231

2232221

1131211

Eliminate

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nnn

n

n

n

b

bbb

a

aaaaaaaaa

......000

..................00...0...

3

2

1

333

22322

1131211

3

Upper triangular matrix

GAUSS ELIMINATION (2)Engineering M

athematics I

1,1

,1111,111,1

nn

nnnnnnnnnnnn

nn

nnnnnn

axab

xbxaxa

abxbxa

kk

n

kjjkjk

k a

xabx

1

Backward substitution

44

EXAMPLE 11323344532

321

321

321

xxxxxxxxx

626071205132

113233445132

Pivot element

* Replace 2nd eq.

(2nd eq.) – 2x(1st eq.)

* Replace 3rd eq.

(3rd eq.) + 1x(1st eq.)

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EXAMPLE 1Engineering M

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1550071202121

626071205132

* Replace 3rd eq.

(3rd eq.) + 3x(2nd eq.)

Upper triangle

325

3

2

1

xxx

POSSIBILITIES (1) Linear equation system has three possibilities of

solutions

Many solutions No solution Unique solution

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1;0;1

330011100111

121311320111

321

xxx

0000

000063304211

632121124211

321

xxx

1000

100063304211

732121124211

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 O

hms

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EXAMPLE #2

P

Q

80V 90V

20 Ohms 10 Ohms

15 Ohms

i1

i2

i3

10 O

hms

8001020902510001110111

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] = 0c1 = 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

150212154244262203

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


b-a

c

b = (1/s)c - (k/s)a


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EXAMPLE 7

i

jk

Linearly independentRank = 3

d e

fd = pe


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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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Linear Equation System