DETEMINANTS. LINEAR SYSTEMS

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ENGINEERING MATHEMATICS II

010.141

Byeng Dong Youn (윤병동윤병동윤병동)윤병동

CHAP. 7

LINEAR ALGEBRA: MATRICS, VECTORS,

DETEMINANTS. LINEAR SYSTEMS

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ABSTRACT OF CHAP. 7

Linear algebra in Chaps. 7 and 8 discusses the theory and application of vectors and matrices, mainly related to

linear systems of equations, eigenvalue problems, and linear transformation.

Chapter 7 concerns mainly systems of linear equations and linear transformations.

Systems of linear equations arise in structural analyses, electrical networks, mechanical frameworks, economic models, optimization problems, numerics (Chaps 21- 23) for differential equations.

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CHAP. 7.1

MATRICS, VECTORS: ADDITION AND SCALAR MULTIPLICATION

A matrix is a mathematical tool to represent linear system characteristics.

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Notations: matrices are denoted by capital boldface letters, such as A; entries at the intersection of the i-row and j-column are denoted by a_ij .

A m × n matrix means m-rows and n-columns; an n × n matrix is called square.

In a square matrix the elements a₁₁ down to a_nn are said to be on the main diagonal.

Vector: a matrix with either one row or one column.

NOTATIONS AND CONCEPTS

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Definition:

MATRIX EQUALITY

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SCALAR MULTIPLICATION

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Problem 1

7A – 5B where

[ ]





−

=

−

=

−

=

26

18 40

26 B

5 A 7

10 40

5 B

5

28 0

21 A

7

2 8 1 B

4 0 3 A

ADDITION AND SCALAR

MULTIPLICATION OF VECTORS

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Problem 2

( )





−

 =

 



−

 



−

=

−

















−

=

















−

=

−

















−

=

















=

−

25 25 25

45 20

20

? C 2D 5

25 45

25 2D

- C 5 5

9 5 D

2 C

6 2 2 D

7 5 9 C

? 2D C

5

ADDITION AND SCALAR

MULTIPLICATION OF VECTORS

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HOMEWORK IN 7.1

HW1. Recall Example 2 (Page 274). Compute the total revenue of the store when three products I, II, III are sold at $100, $120, and $150, respectively.

HW2. Build any matrix to represent your life characteristics. For

instance, grade (% scale) vs. number of study hours per week for three courses (choose any) in the previous semester.

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CHAP. 7.2

MATRIX MULTIPLICATION

The most popular use of matrix multiplication operation is a linear transformation of one system to the other

system.

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Matrix algebraic operation: Addition, Subtraction, Scalar Multiplication, and Matrix Multiplication

Two matrices may be multiplied, and produce a third matrix, provided they are conformable to multiplication.

If there exists A_mn and B_np then these two matrices are

conformable and the resultant matrix is of size ‘mp’, that is, C_mp

MATRIX MULTIPLICATION

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MATRIX MULTIPLICATION DEFINITION

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Example

4 3 7 2 9 0

2 5 1 6

3 2

2 2

















⋅

 

















, 

,

=

8 + 3 20 + 18 14 + 2 35 + 12

18 45

4 2

1 8 = 12 + 10

3 + 40 = 22 43



 

 ⋅

 

 

 

 

 



2 2 2 1 2 1

3

, 5 , ,

MULTIPLICATION EXAMPLES

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Multiplication is not commutative, in general.

AB can be equal to 0, but this does not imply that either A or B is necessarily 0.

Even if it is given that AC = AD, it is not necessarily true that C = D.

Some rules include:

MULTIPLICATION RULES

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Row ×××× Column

Column ×××× Row

[

^{3 6 1}

]

= 3 + 12 + 4

^[ ^]

1 3

3 1

1 1

1 2 4

,

⋅ ,

















[ ]

1 2 4

=

3 6 1

6 12 2

12 24 4

















⋅

















3 1

1 3

3 3

3 6 1

,

ROW AND COLUMN VECTOR

MULTIPLICATION

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MOTIVATION OF MULTIPLICATION

Relation of x₁x₂-system to y₁y₂-system

Further the relation of x₁x₂-system to w₁w₂-system

Finally the relation of w₁w₂-system to y₁y₂-system

C = AB or

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TRANSPOSITION OF MATRICES

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Suppose the matrix A has m rows and n columns; then the transposed matrix A^T has n rows and m columns, with the rows and columns of A interchanged.

 







 







−

 =



 



 −

=

0 1

0 8

4 5

0 ,

0

4 1 8

5

_T

A A

TRANSPOSITION OF MATRICES

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SPECIAL MATRICES

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SPECIAL MATRICES

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If a is a row vector of order n, and b is a column vector, also of order n, then the inner product or dot product of a and b is the scalar which is the sum of the products of the respective elements

Example

Thus, matrix multiplication amounts to combinations of dot products of

[ ]

a = 4

b = 2 5 8

= 8 5 + 40 = 43 (a scalar)

−

















⋅ −

1 5

a b

SPECIAL MATRICES

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MULTIPLICATION OF MATRICES BY

MATRICES AND BT VECTORS

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HOMEWORK IN 7.2

HW1. Problem 19(d)

HW2. Problem 23

HW3. Problem 28(a) and 28(b)

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CHAP. 7.3

LINEAR SYSTEMS OF EQUATIONS.

GAUSS ELIMINATION

The most important use of matrices occurs in the

solution of systems of linear equations, briefly called linear systems.

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LINEAR SYSTEMS OF EQUATIONS

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LINEAR SYSTEMS OF EQUATIONS

In a system of equations there may be a unique solution, infinitely many solutions, or no solution at all.

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EXAMPLE OF LINEAR SYSTEMS

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The Gauss method of elimination is frequently used to solve the linear systems.

Gauss elimination is a standard method for solving linear systems with the following rules:

• Rule 1- rows may be interchanged;

• Rule 2- one row may be multiplied by a non-zero constant

• Rule 3- a multiple of one row may be added to another row

We now call a linear system S₁ row-equivalent to a linear system S₂ if S₁ can be obtained from S₂ by (finitely many)

GAUSS ELIMINATION

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GAUSS ELIMINATION

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GAUSS ELIMINATION

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GAUSS ELIMINATION

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GAUSS ELIMINATION

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GAUSS ELIMINATION

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HOMEWORK IN 7.3

HW1. Problem 7

HW2. Problem 10

HW3. Problem 18

HW4. Problem 22

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CHAP. 7.4

LINEAR INDEPENDENCE. RANK OF A MATRIX. VECTOR SPACE

To answer the questions of existence and uniqueness of solutions, the rank of a matrix and vector space shall be presented.

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LINEAR INDEPENDENCE

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RANK OF A MATRIX

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VECTOR SPACE

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RANK OF A MATRIX

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VECTOR SPACE

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VECTOR SPACE

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p vectors x₍₁₎ … x_(p) (with n components) are linearly independent if the matrix with rows x₍₁₎ … x_(p) has rank p; they are linearly dependent if the rank is less than p.

p vectors with n < p components are always linearly dependent.

The vectors space Rⁿ consisting of all vectors with n components has rank n.

BACKUPS

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Problem 8

the first row is subtracted from row 2 and also from row 3, the result is:

Then subtract row 2 from row 3 and get:

This is sometimes called canonical form, and is obviously of rank 3,

1 0 0 1 1 0 1 1 1

1 0 0 0 1 0 0 1 1

1 0 0 0 1 0 0 0 1

BACKUPS

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Problem 9

2 2

Ra = 1

8 4

2 1

6 3

1

2 1

1

0 0

−

→

−

→

−

Therefore

nk ,

BACKUPS

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Problem 10

3 1 5

2 4 6

10 0 14

1 1

3

5

1 2 33

1 0 7

5

1 1

3

5 3

0 5

3

4 3

0 1

3

7 5

5 3

1 1

3

5 3

0 1 4

5

0 1

3

4 15

1 1

3

5 3

0 1 4

5

0 1 4

5

1 1

3

5 3

0 1 4

0 0 05

− =

=

→

− =

=

→

− =

=  −

 



− =

= −

→

− =

= −

→

=

= −

=

+ +

+

BACKUPS

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HOMEWORK IN 7.4

HW1. Problem 2

HW2. Problem 6

HW3. Problem 22

HW4. Problem 27

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CHAP. 7.5

SOLUTIONS OF LINEAR SYSTEMS:

EXISTENCE, UNIQUENESS

Linear independence (or rank) shall answer the

questions of existence, uniqueness, and general structure of the solution set of linear systems.

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SOLUTION EXISTENCE AND UNIQUENESS

A linear system of equations in n unknowns has a unique solution if the coefficient matrix and the augmented matrix have the same rank n, and infinitely many solution if that common rank is less than n.

The system has no solution if those two matrices have different rank.

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SOLUTION EXISTENCE AND

UNIQUENESS

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HOMOGENEOUS LINEAR SYSTEM

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NONHOMOGENEOUS LINEAR

SYSTEM

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CHAP. 7.7

DETERMINANTS. CRAMER’S RULE

Determinant were originally introduced for solving linear systems but delivers important implications in

eigenvalue problems.

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DETERMINANTS

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DETERMINANTS

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DETERMINANTS

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DETERMINANTS

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CRAMER’S RULE

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( )

x = = 120 + 234 52 + 120

91 =

91 = 2

y = = 26 + 20 39 280

91 = 319 + 46

91 = 273

91 = 3

z = = 26 + 840 60 + 78

91 =

91 = 8 20 2 3

13 3 1

0 6 2

91

354 172 182

1 20 3 7 13 1

1 0 2

91 1 2 20 7 3 13

1 6 0

91

866 138 728

− −

− − − −

−

− −

CRAMER’S RULE

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HOMEWORK IN 7.7

HW1. Problem 11

HW2. Problem 19

HW3. Problem 24 (b)