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Seoul National

Univ. 1

Engineering Math, 7. Linear Algebra

C HAPTER 7. L INEAR A LGEBRA

2019.3 서울대학교 조선해양공학과 서 유 택

※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다.

(2)

Contents

7.1 Matrices, Vectors : Addition and Scalar Multiplication 7.2 Matrix Multiplication

7.3 Linear Systems of Equations. Gauss Elimination 7.4 Linear Independence. Rank of Matrix. Vector Space 7.5 Solutions of Linear Systems : Existence, Uniqueness 7.6 Second- and Third-Order Determinants

7.7 Determinants. Cramer’s Rule

7.8 Inverse of a Matrix. Gauss-Jordan Elimination

(3)

Seoul National

Univ. 3

Engineering Math, 7. Linear Algebra

7.1 M ATRICES , V ECTORS : A DDITION AND S CALAR

M ULTIPLICATION

(4)

Matrices

Matrix : a rectangular array of numbers (or functions) enclosed in brackets.

Entries (성분)

(or sometimes the elements (원소)) Examples)

16 , 2

. 0 0

5 1

3 .

0 

 

,

33 32

31

23 22

21

13 12

11

 

 

a a

a

a a

a

a a

a

 

 

x e

x e

x x

4 2

6

2

rows

columns

square matrices

(no. of rows = no. of columns)

(5)

Seoul National

Univ. 5

Engineering Math, 7. Linear Algebra

Matrices

We shall denote matrices by capital boldface letters A, B, C, ∙∙∙ , or by writing the general entry in brackets ; thus A=[ajk], and so on. By an m x n matrix (read m by n matrix) we mean a matrix with m rows and n columns – rows come

always first!

m x n is called the size of the matrix. Thus an m x n matrix is of the form

).

2 ( ]

[

2 1

2 22

21

1 12

11

 

 

 

 

 

mn m

m

n n

jk

a a

a

a a

a

a a

a a

A

(6)

Matrices

If m = n, we call A an n x n square matrix ( 정방행렬 ).

 

 

 

 

 

mn m

m

n n

jk

a a

a

a a

a

a a

a a

2 1

2 22

21

1 12

11

] [

A

 

 

 

 

 

nn n

n

n n

a a

a

a a

a

a a

a

2 1

2 22

21

1 12

11

A

main diagonal

If m ≠ n, we call A a rectangular matrix ( 구형 또는 장방행렬 ).

(7)

Seoul National

Univ. 7

Engineering Math, 7. Linear Algebra

Vectors

Vector : a matrix with only one row or column.

Examples)

 

 

3 2 1

b b b b

 a

1

a

2

a

3

 ,

 a

row vector column vector

Matrix : a rectangular array of numbers (or functions) enclosed in brackets.

Components (entries)

We shall denote vectors by lowercase boldface letters a, b, ∙∙∙ or by its general component in

brackets, a=[a

i

] , and so on.

(8)

In a system of linear equations, briefly called a linear system, such as

10 8

5

20 2

6

6 9

6 4

3 2

1

3 1

3 2

1

x x

x

x x

x x

x

the coefficients of the unknowns x1, x2, x3 are the entries of the coefficient matrix (계수행렬), call it A,

. 1

8 5

2 0

6

9 6

4

 

 

A

The matrix

 

 

10 20 6

1 8

5

2 0

6

9 6

~ 4 A

is called augmented matrix (첨가행렬) of the system.

We shall discuss this in great detail, beginning in Sec. 7.3 Ex 1) Linear Systems, a Major Application of Matrices

Matrices

(9)

Seoul National

Univ. 9

Engineering Math, 7. Linear Algebra

Matrices

Sales Figures for Three products I, II, III in a store on Monday (M), Tuesday (T), ∙∙∙ may for each week be arrange in a matrix.

. III

II I

780 430

270 0

0 100

960 500

500 780

120 0

470 210

0 810

330 400

 

 

 A 

S F

Th W

T M

If the company has ten stores, we can set up ten such matrices, on for each store. Then by adding corresponding entries of these matrices we can get a matrix showing the total sales of each product on each day.

(10)

Equality of Matrices

Two matrices A=[ajk] and B=[bjk] are equal, written A=B, if and only if they have the same size and the corresponding entries are equal, that is, a11=b11, a12=b12, and so on. Matrices that are not equal are called different. Thus, matrices of different sizes are always different.

Ex3) Equality of Matrices

 

 

 

 

 

 

1 3

0 4

and Let

22 21

12

11

B

A a a a a

. 1 ,

3

, 0 ,

4 if only and

if Then

22 21

12 11

 

a a

a B a

A

(11)

Seoul National

Univ. 11

Engineering Math, 7. Linear Algebra

The sum of two matrices A=[ajk] and B=[bjk] of the same size is written A+B and has the entries ajk+ bjk obtained by adding the corresponding entries of A and B.

Matrices of different sizes cannot be added.

Ex4) Addition of Matrices and Vectors

2 . 2 3

3 5 then 1

0 , 1

3

0 1 5

and 2

1 0

3 6

If 4 

 

 

 

 

 

 

 

  B A B

A

 5 7 2  and  6 2 0  , then  1 9 2  .

If a  b   a  b  

(12)

Scalar Multiplication (Multiplication by a Number)

The product of any m x n matrix A=[ajk] and any scalar c (number c) is written cA and is the m x n matrix cA=[cajk] obtained by multiplying each entry of A by c.

Ex5) Scalar Multiplication

 

 

 

 

5 . 4 0

. 9

9 . 0 0

8 . 1 7

. 2 then

, 5 . 4 0

. 9

9 . 0 0

8 . 1 7

. 2

If A - A

0 0

0 0

0 0 0

, 5 10

1 0

2 3

9 10

 

 

 

 

 A

A

zero matrix

(13)

Seoul National

Univ. 13

Engineering Math, 7. Linear Algebra

Rules for Matrix Addition and Scalar Multiplication

 

 

0 )

( )

(

0 )

(

) (

) (

) (

) ( )

3 (

A A

A A

C B

A C

B A

A B

B A

d c b a

 

 

A A

A A

A A

A

B A

B A

1 )

(

) (

) (

) (

) (

) (

) (

) ( ) 4 (

d

ck k

c c

k c

k c

b

c c

c a

 written A  B  C 

 written ck A 

0: zero matrix (of size m x nmatrix)

(14)

7.2 M ATRIX M ULTIPLICATION

(15)

Seoul National

Univ. 15

Engineering Math, 7. Linear Algebra

) 1

2

(

2 1

1 1

jn nk

k j

k j n

l

lk jl

jk

a b a b a b a b

c      

1 , , .

, , 1

p k

m j

C B

A

] [

] [

]

[mn nr mr

 

 

 

 

43 42

41

33 32

31

23 22

21

13 12

11

a a

a

a a

a

a a

a

a a

a

11 12

21 22

31 32

( 3)

b b

b b r

b b

  

    

 

  

 

11 12

21 22

31 32

41 42

4 c c

c c c c m c c

  

  

  

        

 4 m

 3

n p  2 p  2

31 23 21

22 11

21

21

a b a b a b

c   

Multiplication of a Matrix by a Matrix

(16)

Multiplication of a Matrix by a Matrix

 

 

 

 

1 1

4 9

8 7

0 5

1 3

2 2

2 3

6

2 0

4

1 5

3 AB

 

 

28 37

4 9

6 14

16 26

42 43

2 22

22 9

) 1 ( 5 5 2

3       

c11

14 1

2 7

0 3

4      

c23

The product BA is not defined.

Ex 1) Matrix Multiplication

C B

A

] [

] [

]

[mn nr mr

(17)

Seoul National

Univ. 17

Engineering Math, 7. Linear Algebra

 

 

 

 

5 3 8

1

2

4 

 

 

5 8 3

1

5 2 3

4 

 

 

43 22

 

 

 

 

8 1

2 4

5

3

is not defined.

Ex 1) Matrix Multiplication

Multiplication of a Matrix by a Matrix

[mAn] [nBr] [mCr]

 

 

 

 4 2 1 1

6

3    19

Ex3) Products of Row and Column Vectors

 3 6 1 

4 2 1

 

 

. 4 24

12

2 12

6

1 6

3

 

 

(18)

 

 

 

 

 

 

1 1

1 , 1

100 100

1

1 B

A

 

 

 

 

 

1 1

1 1

100 100

1

AB 1 

 

 

0 0

0 0

 

 

 

 

 

100 100

1 1

1 1

1

BA 1 

 

 

99 99

99 99

BA AB 

It is interesting that this also shows that AB=0does notnecessarily imply BA=0 or A=0or B=0.

Ex 4) CAUTION! Matrix Multiplication is Not Commutative (가환성의), AB≠BA in General

Multiplication of a Matrix by a Matrix

[mAn] [nBr] [mCr]

Commutative rule (교환법칙)

(19)

Seoul National

Univ. 19

Engineering Math, 7. Linear Algebra

Properties of Matrix Multiplication

( ) ( ) ( ) ( )

( ) ( ) ( )

(2) ( ) ( )

( ) ( )

a k k k

b c d

 

  

    

    

A B AB A B A BC AB C

A B C AC BC C A B CA CB

 written k AB or A k B 

 written ABC 

(2b) is called the associative law (결합법칙)

분배법칙

)
(20)

) 1

2

(

2 1

1 1

jn nk

k j

k j n

l

lk jl

jk

a b a b a b a b

c      

Since matrix multiplication is a multiplication of rows into columns, we can write the defining formula (1) more compactly as

) 3

 (

k j

c

jk

 a b j  1 ,  , m ; k  1 ,  , p .

where

a

j : the jth row vector of

A

b

k : the kth column vector of

B

, so that in agreement with (1),

 

j k j k jn nk

nk k jn

j j

k

j

a b a b a b

b b a

a

a    

 

 

  

1 1 2 2

1 2

b

1

a

C B

A

] [

] [

]

[mn nr mr

Properties of Matrix Multiplication

(21)

Seoul National

Univ. 21

Engineering Math, 7. Linear Algebra

If

A=[a

jk

]

is of size 3x3 and

B=[b

jk

]

is of size 3x4, then

1 1 1 2 1 3 1 4

2 1 2 2 2 3 2 4

3 1 3 2 3 3 3 4

(4).

 

 

  

 

 

a b a b a b a b AB a b a b a b a b a b a b a b a b

Parallel processing of products on the computer is facilitated by a variant of (3) for computing C=AB, which is used by standard algorithms.

In this method, A is used as given, B is taken in terms of its column vectors, and the product is computed columnwise; thus,

1 2 p 1 2 p

(5).

   

     

AB A b b b Ab Ab Ab

1 2 3 4

3 2 1

, B b b b b

a a a

A





Ex 5) Product in Terms of Row and Column Vectors

Properties of Matrix Multiplication

(22)

1 2 p 1 2 p

(5).

   

     

AB A b b b Ab Ab Ab

Columns of B are then assigned to different processors (individually or several to each processor), which simultaneously compute the

columns of the product matrix Ab

1

, Ab

2

, etc.

1 1 1 2 1 3 1 4

2 1 2 2 2 3 2 4

3 1 3 2 3 3 3 4

(4).

 

 

  

 

 

a b a b a b a b AB a b a b a b a b a b a b a b a b

jk j k

c a b

Properties of Matrix Multiplication

(23)

Seoul National

Univ. 23

Engineering Math, 7. Linear Algebra

To obtain AB from (5), calculate the columns

 

 

 

 

 

 

 

 

23 8

17

34 4

11 6

4 1

7 0

3 2

5

1 AB 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 23

34 6

7 2

5

1 , 4

8 4 4

0 2

5

1 , 4

17 11 1

3 2

5

1 4

Ex 6) Computing Products Columnwise by (5)

Properties of Matrix Multiplication

1 2 p 1 2 p

(5).

   

     

AB A b b b Ab Ab Ab

(24)

Applications of Matrix Multiplication

Ex 11) Computer Production. Matrix Times Matrix

Supercomp Ltd produces two computer models PC1086 and PC1186.

Matrix A : the cost per computer (in thousands of dollars)

Matrix B : the production figures for the year 2010 (in multiples of 10,000 units) Find a matrix C that shows the shareholders the cost per quarter (in millions of dollars) for raw material, labor, and miscellaneous.

PC1086 PC1186

1.2 1.6 0.3 0.4 0.5 0.6

 

 

  

 

 

A

Raw Components Labor

Miscellaneous

 

 

 

3 4

2 6

9 6

8 B 3

PC1086 PC1186 Quarter

1 2 3 4

(25)

Seoul National

Univ. 25

Engineering Math, 7. Linear Algebra

Applications of Matrix Multiplication

Ex 11) Computer Production. Matrix Times Matrix

1.2 1.6 0.3 0.4 0.5 0.6

 

 

  

 

 

A

PC1086 PC1186

Miscellaneous

 

 

 

3 4

2 6

9 6

8

B 3

PC1086

PC1186 Quarter

1 2 3 4

 

 

3 . 6 4

. 5 2

. 5 1

. 5

9 . 3 4

. 3 2

. 3 3

. 3

6 . 15 6

. 13 8

. 12 2

. 13 AB

C

Quarter

1 2 3 4

Raw Components Labor

Miscellaneous

Q ?

(26)

A weight-watching program

A person of 185lb burns 350 cal/hr in walking (3 mph) 500 cal/hr in bicycling (13 mph)

950 cal/hr in jogging (5.5 mph)

Bill, weighing 185 lb, plans to exercise according to the matrix shown.

Verify the calculations (W = Walking, B = Bicycling, J=Jogging).

 

 

 

 

 

 

 

 

 

 

2400 1000 1325 825

950 500 350

0 . 1 5

. 1 0

. 2

5 . 0 0

5 . 1

5 . 0 0

. 1 0

. 1

5 . 0 0

0 . 1

W B J MON

WED FRI SAT

MON WED FRI SAT

Ex 12) Weight Watching. Matrix Times Vector

Applications of Matrix Multiplication

(27)

Seoul National

Univ. 27

Engineering Math, 7. Linear Algebra

Applications of Matrix Multiplication

Suppose that the 2004 state of land use in a city of 60 mi2 of built-up area is C : Commercially Used 25 %,

I : Industrially Used 20 %, R : Residentially Used 55 %.

Transition probabilities for 5-year intervals : A and remain practically the same over the time considered.

Find the states 2009, 2014, 2019

0.7 0.1 0

: 0.2 0.9 0.2

0.1 0 0.8

 

 

 

 

 

A

From C From I From R To C

To I To R

Ex 13) Markov Process. Powers of a Matrix. Stochastic Matrix

* Markov Process: 미래 상태가 과거 상태와는 독립적으로 현재 상태에 의해서만 결정된다는 것

(28)

확률행렬

) : a square matrix

all entries nonnegative all column sums equal to 1

Markov process : the probability of entering a certain state depends only on the last state occupied (and the matrix A), not on any earlier state.

solution.

 

 

55 8

. 0 20

0 25

1 . 0

55 2

. 0 20

9 . 0 25

2 . 0

55 0

20 1

. 0 25

7 . 0

area C

I R

. 5 . 46

0 . 34

5 . 19

55 20 25

8 . 0 0

1 . 0

2 . 0 9

. 0 2

. 0

0 1

. 0 7

. 0

 

 

 

 

 

 

Ex 13) Markov Process. Powers of a Matrix. Stochastic Matrix

Applications of Matrix Multiplication

0.7 0.1 0

: 0.2 0.9 0.2

0.1 0 0.8

 

 

 

 

 

A

From C From I From R To C

To I To R

(29)

Seoul National

Univ. 29

Engineering Math, 7. Linear Algebra

Applications of Matrix Multiplication

 

 

55 8 . 0 20 0 25

1 . 0

55 2 . 0 20 9 . 0 25 2 . 0

55 0 20

1 . 0 25 7 . 0

area C

I R

. 5 . 46

0 . 34

5 . 19

55 20 25

8 . 0 0

1 . 0

2 . 0 9 . 0 2 . 0

0 1

. 0 7 . 0

 

 

 

 

 

 

We see that the 2009 state vector (y) is the column vector

 

 

 

 

55 20 25

5 . 46

0 . 34

5 . 19

A Ax

y

The vector (x) is the given 2004 state

vector

The sum of the entries of y is 100%.

Similarly, you may verify that for 2014 and 2019 we get the state vectors.

 

T

2

17 . 05 43 . 80 39 . 15 )

(  

 Ay A Ax A x z

 

T

3

16 . 315 50 . 660 33 . 025 )

(  

 Az A Ay A x u

0.7 0.1 0

: 0.2 0.9 0.2

0.1 0 0.8

A

From C From I From R To C

To I To R

Q ?

(30)

Transposition (전치)

Transposition of Matrices and Vectors

The transposition of an m x n matrix A=[ajk] is the n x m matrix AT (read A transpose)

: the first row of A as its first column

the second row of A as its second column, and so on.

Thus the transpose of A in (2) is

A

T

=[a

kj] , written out

As a special case, transposition converts row vectors to column vectors and conversely

mn m

m

n n

a a

a

a a

a

a a

a

2 1

2 22

21

1 12

11

A

) 9 ( ]

[

2 1

2 22

12

1 21

11

T

 

 

 

 

 

mn n

n

m m

kj

a a

a

a a

a

a a

a a

A

(31)

Seoul National

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Engineering Math, 7. Linear Algebra

 

 

 

 

 

0 1

0 8 -

4 5

then 0 ,

0 4

1 8 5

If A A

T

, 0 1

0 8 -

4 5

0 0

4

1 8

5

T

 

 

 

 

 

1 , - 0

8 3

1 - 8

3

T

 

 

 

 

 

  ,

3 2 6 3

2

6

T

 

 

  6 2 3 

3 2 6

T

 

 

Ex 7) Transposition of Matrices and Vectors

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Rules for transposition

T T

T T

T T

T T T

T

( ) ( )

( ) ( )

( ) ( )

( ) ( )

a b

c c c

d

 

   

 

 

  A

A A

A B A

B B

B

A A

A

Caution! Note that in (d) the transposed matrices are in reversed order.

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)

반대칭행렬

)

)

thus (

T

,

jk

kj

a

a 

 A A

).

0

hence ,

thus (

T

  A , a

kj

  a

jk

a

jj

A

Ex 8) Symmetric and Skew-symmetric Matrices

 

 

30 150

200

150 10

120

200 120

20 A

 

 

0 2

3

2 0

1

3 1

0 B

Symmetric matrix Skew-symmetric matrix

0 0

0

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Triangular Matrices

Ex 9) Upper and Lower Triangular Matrices 2 ,

3 1 

 

0 ,

6 0

2 3

2 4

1

 

 

 0

0 0

, 8 6

7

0 1

8

0 0

2

 

 

 0

0 0

. 6 3

9 1

0 2

0 1

0 0

3 9

0 0

0 3

 

 

 

 

 0

0 0

0 0

0

Upper Triangular Matrices Lower Triangular Matrices

위삼각행렬

)

square matrices

nonzero entries only on and above the main diagonal

any entry below the diagonal must be zero.

아래삼각행렬

)

nonzero entries only on and below the main diagonal

Any entry on the main diagonal of a triangular matrix may be zero or not.

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Engineering Math, 7. Linear Algebra

Diagonal Matrices

Ex 10) Diagonal Matrix D. Scalar Matrix S. Unit Matrix I.

, 0 0

0 3

0 0

2

 

 

 D

Diagonal Matrices ( 대각행렬 )

 square matrices

 nonzero entries only on the main diagonal

 any entry above or below the main diagonal must be zero.

, 0

0 0

0 0

 

 

c c

c

S .

1 0

0 1

0

0 0

1

 

 

I

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Diagonal Matrices

, 0 0

0 3

0 0

2

 

 

D ,

0 0

0 0

0 0

 

 

c c

c

S .

1 0

0 1

0

0 0

1

 

 

 I

S : scalar matrix ( 스칼라 행렬 )

).

12

 ( A SA

AS   c

I : unit matrix (or identity matrix) ( 단위행렬 )

a scalar matrix whose entries on the main diagonal are all 1

).

13

 ( A IA

AI  

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R EFERENCE : V ECTORS AND L INEAR E QUATIONS *

*Strang G., Introduction to Linear Algebra, Third edition, Wellesley-Cambridge Press, 2003, Ch.2.1, p21

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Vectors and Linear Equations

The import problem of linear algebra is to solve a system of equations.

2 1

3 2 11

x y

x y

 

 

First example)

Slopes are important in calculus and this is linear algebra.

1 2 

 y x

11 2

3 x  y 

1 1

3 y

x

3 11

The point x = 3, y = 1 lies on both lines.

This is the solution to our system of linear equations.

solution of first equation solution of second equation

The left figure shows two lines meeting

at a single point.

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Vectors and Linear Equations

11 2

3

1 2

y x

y x

Linear system ( 선형연립방정식 ) can be a “vector equation”. If you

separate the original system into its columns instead of its rows, you get

 b

 

 

 

 

 

 

 

 

11 1 2

2 3

1 y

x

y

x



 

 3

 1

 



2 2



 

 11

1



 

 9 3

A combination of columns produces the right side (1, 11).

This has two column vectors on the left side. The problem is to find the combination of those vectors that equals the vector on the right.

We are multiplying the first column by x and the second column by y, and adding. With the right choices x = 3, y = 1, this produces 3

(column 1) + 1 (column 2) = b.

The left figure combines the column vectors on the left side to produce the vector b on the right side.

1 2

y x

11 2

3x y

1 1

3 y

x

3 11 비교

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Vectors and Linear Equations

3 2 11 1 2

y x

y x

 b

 

 

 

 

 

 

 

 

11 1 2

2 3

1 y

x

The left side of the vector equation is a linear combination of the columns.

The problem is to find the right coefficients x = 3 and y = 1.

We are combining scalar multiplication and vector addition into one step. :

1 ,

3 

 x y

1 2 1

3 1

3 2 11

      

 

     

     

Linear combination

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Vectors and Linear Equations

- Three Equations in Three Unknowns

The three unknowns x, y, z. The linear equations Ax = b are

2 3 6

2 5 2 4

6 3 2

x y z

x y z

x y z

  

  

  

The right figure shows three planes meeting at a single point.

x

y z

6 3 2

y z x

4 2 5

2x y z

x

y z

2 3

6x y z

L L

The usual result of two equations in three unknowns is a intersect line L of solutions.

The third equation gives a third plane. It cuts the line L at a single point. That point lies on all three planes and it satisfies all three

equations.





 2 0 0

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Vectors and Linear Equations

- Three Equations in Three Unknowns

1 2 3 6

2 5 2 4

6 3 1 2

x y z

       

          

       

        

       

The left figure combines three columns to produce the vector (6,4,2)

x

y z

1 column 6

2 1

2 column 3

5 2

1 2 3

3 column

times 2

2 4 6

b

2 3 6

2 5 2 4

6 3 2

x y z

x y z

x y z

The left figure starts with the vector form of the equations :

(x, y, z) = (0, 0, 2) because 2(3, 2, 1) = (6, 4, 2) = b.

The coefficient we need are x = 0, y = 0 and z = 2. This is also the intersection point of the three planes in

the right figure.

x

y z

2 3

6x y z

L





 2 0 0

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Vectors and Linear Equations

- The Matrix Form of the Equations

b

Ax 

 

 

 

 

 

 

2 4 6

1 3

6

2 5

2

3 2

1

z y x

) 3 ( 2

3 6

4 2

5 2

6 3

2

z y x

z y

x

z y x

Coefficient matrix unknown vector Matrix equation :

We multiply the matrix A times the unknown vector x to get the right side b.

Multiplication by rows : Ax

comes from dot products, each row times the column x :

. )

(

) (

) (

 

 

x 3

row

x 2

row

x 1

row Ax

Multiplication by columns : Ax is a combination of column

vectors :

) (

) (

) (

3 column 2

column 1 column Ax

z y

x

(44)

Vectors and Linear Equations

- The Matrix Form of the Equations

Ax b

2 4 6 1

3 6

2 5 2

3 2 1

z y x

When we substitute the solution x = (0, 0, 2), the multiplication Ax produces b :

) (

) (

)

( column 1 column 2 column 3

Ax  x  y  z

. 2 4 6 3

column times

2 2

0 0

1 3

6

2 5

2

3 2

1

 

 

 

 

 

 

The first dot product in row multiplication is (1, 2, 3) • (0, 0, 2) = 6. The other dot products are 4 and 2.

Multiplication by columns is simply 2 times column 3.

Ax is a combination of the columns of A.

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7.3 L INEAR S YSTEMS OF E QUATIONS . G AUSS

E LIMINATION

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coefficients

If b1=∙∙∙=bm

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