chapter 3 linear algebra

Chapter 3 Linear Algebra

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture 7 Matrix

1 Introduction

- Algebra & Geometry

- Vector

- Change of coordinates (transformation)

H. W. (Due Apr. 16th)

Chapter 3

3-2. 8, 9

3-3. 1

3-5. 12, 16, 21, 37

3-6. 6, 15

3-7. 23, 24

“The problems similar to the above ones will appear in the midterm exam.”

2. Matrix; row reduction ( 행렬 ; 행줄이기 )

- A matrix is just a rectangular array of quantities, usually enclosed in large parentheses.

.6,0,3,2,5,1

number)(column number), (row:

matrix 2)by 2(32:603

251

232221131211

AAAAAA

jiAij

A

- Transpose of a matrix

jiij A

T

T

A

A matrix23:

62

05

31

- Sets of Linear Equations

42

7356

22

yx

zyx

zx

.3,2,1,

.

4

7

2

,,

012

356

102

where

,

4

7

2

012

356

102

,

3

1

ikxM

z

y

x

z

y

x

ij

jij

krM

kMr

4012

7356

2102

A

2110

1650

2102

2

165

22

zy

zy

zx

- Augmented matrix

(a) Eliminate the x terms in the other two equations by using the first equation.

ex. 2) – 1) x 3 2) , 3) – 1) 3)

(b) For convenience, interchange the second and third equations

1650

2110

2102

165

2

22

zy

zy

zx

42

7356

22

yx

zyx

zx

(c) Eliminate the y terms by using the second equation.

ex. 3) – 2) x 5 3)

111100

2110

2102

1111

2

22

z

zy

zx

(d) Eliminate the z terms by using the third equation.

ex. 1) + 3) / 11 1) , 2) – 3) / 11 2)

111100

1010

3002

1111

1

32

z

y

x

(e) finalizing

1100

1010

2/3001

1

1

32

z

y

x

- Allowed rules

i. Interchange two rows

ii. Multiply (or divide) a row by a (nonzero) constant

iii. Add a multiple of one row to another; this includes subtracting, that is,

using a negative multiple.

Ex. 2

942

4832

54

zyx

zyx

zyx

20000

6010

5411

4010

6010

5411

9421

4832

5411row 2nd with therow1st with the

We can not get an answer. The equations are inconsistent.

- Rank of a Matrix

- The number of nonzero rows remaining when a matrix has been row reduced is called the rank of a matrix.

942

4832

54

zyx

zyx

zyx

20000

6010

5411

000

010

411

,

20000

6010

5411

MA

rank: 3 rank: 2 ‘Inconsistent’

- For example 2, the rank of A is 3, but the rank of M is 2. In this case, the equations are inconsistent [(rank of M) < (rank of A)].

a. If (rank M) < (rank A), the equations are inconsistent and there is no solution.

b. If (rank M) = (rank A) = n (number of unknowns), there is one solution.

c. If (rank M) = (rank A) < n , then R unknown can be found in terms of the

remaining n – R unknowns.

3. Determinants; Cramer’s rule ( 행렬식 ; Cramer 의 규칙 )

- We have said that a matrix is simply a display of a set of numbers; it does not

have numerical value. For a square matrix, however, there is a useful number

called the determinant of the matrix.

- Evaluating determinants

.det, bcaddc

ba

dc

ba

AA

i) 2 by 2

nnnnn

n

n

n

aaaa

aaaa

aaaa

aaaa

321

3333231

2232221

1131211

- When removing the row and the column containing the element a_ij, we have the remaining determinant, M_ij, called a minor of a_ij.

ii) nth order matrix

321

3333231

2232221

1131211

33

nnnnn

n

n

n

aaaa

aaaa

aaaa

aaaa

M

ex)

(n-1) by (n-1) determinant

- Finally, multiply each element of one row (or one column) by its cofactor and add the results.

)(

1detjori

ijijji MaA

etc

etc

- Sign

- cofactor : ijji M 1

“Laplace’s development”

12

51,4

512

437

251

2323

Ma

.1483851141237

515

12

514

12

372

512

437

251

.14821351112

372

52

475

51

431

512

437

251

ex)

i) method 1 (third column)

ii) method 2 (first row)

- Useful facts about determinants

1. If each element of one row (or one column) of a determinant is multiplied by a

number k, the value of the determinant is multiplied by k.

2. The value of a determinant is zero if,

(a) all elements of one row are zero

(b) two rows ( or two columns) are identical

(c) two rows (or two columns) are proportional.

3. If two rows ( or two columns) of a determinant are interchanged, the value of

the determinant changes sign.

4. The value of a determinant is unchanged if

(a) rows are written as columns are columns as rows

(b) we add to each element of one row, k times the corresponding element of

another row, where k is any number (and a similar statement for columns).

Example 2. Find the equation of a plane through the three given points (0,0,0),

(1,2,5), and (2,-1,0)

0

1012

1521

1000

1

zyx

Example 4. Evaluate the determinant

0413

1204

3279

1034

D

3643

20611

060

473

230

1

413

473

230

1

0413

1000

3473

1230

1. Subtract 4 times the fourth column from the first column, and subtract 2 times the

fourth column from the third column

2. Do a Laplace development using the third row

3. Add the second row to the third row.

4. Do a Laplace development using the third row.

- Cramer’s rule

.,

22

11

22

11

22

11

22

11

2

1

22

11

222

111

ba

ba

ca

ca

y

ba

ba

bc

bc

x

c

c

y

x

ba

ba

cybxa

cybxa

kMr

- We can use this method when you get the solution for the n linear equations

(in case that D 0).

- Denominator: determinant of the matrix with coefficients in the left side (M)- Numerator: for x, replace x part in M with right side part, and take the determinant.

for y, replace y part in M with right side part, and take the determinant.

- Rank of a Matrix

To find the rank of a matrix, we look at all the square submatrices and find their

determinants. The order of the largest nonzero determinant is the rank of the

matrix.

cf. submatrix: a matrix remaining if we remove some rows and/or columns.Example 6.

6544

0122

3211

The submatrices of the original matrix are four (123, 124, 134, and 234).

Because the first and the second columns are absolutely the same, the determinants

of 123 and 123 should be zero. The absolute values of 134 and 234 should be the

same. For this reason, we need to check only 134.

0

654

012

321

6544

0122

3211submatrix

“The rank should be less than 3.”



Lecture 8 Vector

4. Vector ( 벡터 )

- Notation

yx AAA ,

A

- Magnitude 222zyx AAAA A

- Addition of vectors

addtionfor law eassociativ:

additionfor law ecommutativ:

CBACBA

ABBA

- Multiplication by a constant & subtraction

- Unit vector A

A

- Vectors in terms of components

zzyyxx

zyx

BABABA

AAA

kjiBA

kjiA

- Multiplication of vectors 1: scalar product

ABBA

BABA

cos

zzyyxx BABABA

A

BA

CABACBA

AA

BAB

ABABA

2

on of projectiontimes

on of projectiontimes

- Angles between two vectors using scalar product

BA

BAcos

example) Find the angles between these two vectors

1,3,2,9,6,3 BA

60,2

1

14143

21cos

14,143963

21193623

222

BA

BA

BA

BA

- Perpendicular and Parallel vectors

z

z

y

y

x

x

B

A

B

A

B

Ac

BABA

BABA

//

0

- Multiplication of vectors 2 : vector product

BABABABA

BA

,,sin

productvector :

ABBA

AA

BABA

0

//0

.,,,

0

kijjikikjkji

kkjjii

zyx

zyxzyxzyx

BBB

AAABBBAAA

kji

kjikjiBA

example 4. 2,3,1,1,1,2 BA

.53

231

112 kji

kji

BA

5. Lines and planes ( 직선과 평면 )

- A great deal of analytic geometry can be simplified by the use of vector notation.

Such things as equations of lines and planes, and distances between points or

between lines and planes often occur in physics, and it is very useful to be able to

find them quickly. Vector notation will help you do these more easily.

‘points vectors’

To determine a specific line, we need one point and a slope (= two points).

cba

zzyyxx

,,

,, Slope cf. 121212

A

- Straight Lines

zyx

zyx

,, Variable

,,pointGiven 1110

r

r

.0,,

)0,, line,straight a ofequation symmetric(,

line)straight a of equations c(parametri

,

,

,

,

000

000

0

0

0

00

czzb

yy

a

xx

cbatc

zz

b

yy

a

xx

ctzz

btyy

atxx

tort ArrArr

To determine a specific plane, we need a point and a normal vector.

plane) a of (equation ,0

0

000

0

dczbyaxorzzcyybxxa

rrN

- Planes

Example 1. Find the equation of the plane through the three points A(-1,1,1),

B(2,3,0), C(0,1,-2).

- Because points are given, what we have to do is to find the normal vector.

AB

C

N

AB

AC

0023826 :plane theofEquation

.286

301

123

3,0,1),1,2,3()0,1,1(0,3,2

zyx

ACAB

ACAB

kji

kji

N

Example 1. Find the equation of the plane through the three points A(-1,1,1),

B(2,3,0), C(0,1,-2).

Example 2 Find the equation of a line through (1,0,-2) and perpendicular to the

plane of Example 1

)1

2

43

1(

2

2

86

1 : line theofequation

2,8,6

zyxor

zyx

A

Example 3 Find the distance from the point P(1,-2,3) to the plane 3x-2y+z+1=0.(Use the dot product.)

P

RQ PR: distance we want to know

Q : any point on the plane

product.)dot the torelated is projection (The cosPQPR

- Choose Q in the easiest way, e.g., Q=(0,0.-1) or (1,2,0) (as in the text)

.14/1114/1,2,34,2,1

14,1,2,3,4,2,13,2,11,0,0

PR

PQ NN

.for ,

//

NNnnNN

N

PQPQPR

PR

Example 4 Find the distance from P(1,2,-1) to the line joining P1(0,0,0) and

P2(-1,0,2). (Use the cross product.)

./ where,sin AAaa PQPQPR

.5215/1,2,1)2,0,1(

1,2,1,2,0,1

1

121

PPPR

PPPQPP

a

A

Example 5 Find the distance between the lines, r=i-2j+(i-k)t, r=2j-k+(j-i)t.

A

B

P

Q

n nBABAn PQ/

321,1,1,1,4,1

0,1,1,1,2,0

1,0,1),0,2,1(

nBA

B

A

PQPQ

Q

P

Example 6. Find the direction of the lines of intersection of the planes

Note) the intersection lines are perpendicular to both the planes.

Example 7. Find the cosine of the angle between two planes

Note) The angle between the planes is the same as the angle between the normal vectors to the planes



Lecture 9 Matrix operation

6. Matrix operations ( 행렬계산 )

- Matrix equations

- Multiplication of a matrix by a number

.1,5,7,1,3,2

173

512

ivuisryx

iivsy

urx

kfkdkb

kekcka

fdb

ecak

T 32or3

232 AAjiA

- Addition of Matrices

Note) Addition (or subtraction) can be done with the same type of matrices

?53

12

174

231cf.

107

223

217734

421321

273

412

174

231

- Multiplication of matrices

The element in row i and column j of the product matrix AB is equal to row

i of A times column j of B. [(# of row of A) = (# of column j)]

In index notation,

k

kjikij BAAB

CAB

dhbfdgbe

chafcgae

hg

fe

db

caex.

Example 1

413371532113

423472542214

472

351

13

24

472

351,

13

24

AB

BA

Example 2. Find AB and BA

1333

17,

234

322

37

25,

24

13

BAAB

BA

“not commutative”

- Zero matrix

: zero or null matrix means one with all its elements equal to zero.

00

00

21

42 2MMcf.

- Identity matrix or Unit matrix

AAIIAI

,

100

010

001

- Operation with Determinants

BABAAB detdetdetdet

- Applications of matrix multiplication

kMrkMr 1Then,

10

1

5

231

032

101

2 Method

10

1

5

23

321 Method

10

1

5

231

032

101

z

y

x

zyx

yx

zx

z

y

x

- Inverse of a Matrix

IMMMM 1 1

ijji

ijijT MmC 1, ofcofactor where

det

11 CM

M

Example 3. 3det,

231

032

101

MM

.332

01 ,2

02

11 ,3

03

10 : row 3rd

,331

01 ,3

21

11 ,3

23

10 : row 2nd

,331

32 ,4

21

02 ,6

23

03 : rowst 1

333

234

336

3

1

det

1so

323

333

3461- TC

MMC

Finding the cofactor

- Rotational matrices

cossin

sincos

cossin

sincos

cossin

sincos



Lecture 10 Linear transformation

7. Linear combinations, linear functions, linear operators ( 선형결합 , 선형함수 , 선형연산자 )

- A function of a vector, say f(r), is called linear if

rrrrrr afaffff and,2121

21212121

21212121

2 Ex.

.

.

),1,3,2( 1 Ex.

rrrrrrrr

rr

rrArAr

rrrArArrArr

rArA

fff

f

afaaaf

fff

f

linear

not linear

- F(r) is a linear vector function if

rFrFrFrFrrF aa and,2121

- Linear operator

AABABA kOkOOOO and

- Matrix operators, Linear transformations

MrR

or,or

,

,

y

x

dc

ba

Y

X

dycxY

byaxX

Moving a point to some other point

mapping or transformation

M : transformation matrix (linear operator)

ii) two sets of coordinates (x,y) (x’,y’) & one vector r = r’

Mrr

or,or

,

,

y

x

dc

ba

y

x

dycxy

byaxx

i) One set coordinate & r R

- Orthogonal transformation

: linear transformation preserves the length of a vector.

T

yxyx

MM

1

2222

:maxtrix Orthogonal

:mation transforOrthogonal

cf. det M = 1 : rotation, det M = -1 : reflection

10

01

0,1,1

2

22

221

2222

222222222222

dbcdab

cdabca

dc

ba

db

ca

cdabdbca

yxydbxycdabxcadcxbaxyx

T MMMM

prove)

1det

detdet1detdetdetdetdet 2

M

MMMMMIMM TTT

Example 3. Find what transformation correspond to each of these matrices

.,,10

01,

13

31

2

1BADABCBA

reflectiondetdetdet,1detdetdet,1det

rotation,1det

ABDBACB

A

rotation 12032

1or

3

1

2

1

0

1

13

31

2

1

jii

i) A

ii) B : y -y, reflection through the x axis.

iii) C

,13

31

2

1

10

01

13

31

2

1

ABC

- We want to find the reflection line. The vector lying on the reflection line is not changed by the reflection.

3say,,3

13

31

2

1

ji

xy

y

x

y

x

3

1

1

3

1

3

1

3

13

31

2

1

3

1

3

1

13

31

2

1

iv) D

13

31

2

1

13

31

2

1

10

01BAD

Analyzing the D in the same way,

3say,,3

13

31

2

1

ji

xy

y

x

y

x

- Rotation in 2 Dimensions

y

x

Y

X

cossin

sincos

y

x

y

x

cossin

sincos

vector rotated(active)

axes rotated(passive)

change of basis

kjikji ,,,,

- Rotations and Reflections in 3 Dimensions

100

0cossin

0sincos

A

100

0cossin

0sincos

B

cos0sin

010

sin0cos

F

rotating along the z-axis

rotating along the z-axis + reflection through xy plane

rotation along the y axis

8. Linear dependence and independence ( 선형종속과 선형독립 )

Three vectors A=i+j, B=i+k, C=2i+j+k are linearly dependent,

because A+B-C=0.

0

112

101

011

ly,Equivalent

- Linear independent of functions

0332211 xfkxfkxfkxfk nn

In this case, these functions are linearly dependent.

ex. 1)

ex. 2)

xx 22 cos1,sin

xx cos,sin

linearly dependent

linearly independent

t.independenlinearly are functions thethen,

,0

'

wronskian

if and,1order of derivative have ,,, If

111

1

321

321

21

xfxf

xf

xfxfxf

xfxfxfxf

W

nxfxfxf

nn

n

n

n

Example 1. xx sin,,1

0sin

sin00

cos10

sin1

x

x

x

xx

W

Example 2. xxxx sin32,sin,

0

sin30sin0

cos32cos1

sin32sin

xx

xx

xxxx

W

- Homogeneous equations (right sides are zero)

022

011

022

0

010

001

0

0

yx

yx

yx

yx x=y=0, (rank 2) = (# of unknowns)

(trivial solution)

all points on x+y=0, (rank 1) < (# of unknowns)

(nontrivial solution)

Example 4. For what values of does the set of eqs. have nontrivial solutions?

5,0,042

21

042

021

yx

yx

9. Special matrices and formulas ( 특별한 행렬과 공식들 )

- Summary “Please take a look at (9.1) and (9.2)

- Index notation

k

kjikij BAAB

- Kronecker

jiif

jiifij

,1

,0

ij

ij

dxmxnx

nmif

nmifdxmxnx

coscos

,0

,coscos

100

010

001

I

- More useful theorems

11111

ABCDABCD

ABCDABCD TTTTT



Lecture 11 Diagonalizing matrices

10. Linear vector space ( 선형 벡터공간 )

(Please read this section individually.)

MrR

or,or

,

,

y

x

dc

ba

Y

X

dycxY

byaxX

Moving a point to some other point

mapping, transformation, or deformation

M : transformation matrix (linear operator)

- One set coordinate & r R

11. Eigenvalues and eigenvectors; diagonalizing matrices ( 고유값과 고유벡터 ; 행렬의 대각화 )

Some vectors are not changed in direction by transformation.

const. where rRMrR

r : eigenvectors (characteristic vector): eigenvalues

- Definition of eigenvalue and eigenvector

- Eigenvalues ( 고유값 )

.22

25

y

x

Y

X

.0)2(2

,02)5(or

,22

,25

:condition rEigenvecto22

25

yx

yx-

yyx

μxyx-

y

x

y

x

y

x

Y

X

rR

seigenvalue.6or 1

,0674)2)(5(

.matrix ofequation sticcharacteri 022

25

,0 other thansolution afor Condition

2

M

yx

According to the definition,

- Eigenvectors ( 고유벡터 )

- From the above results,

6.for 02

1for 02

yx

yx

.22

25 through

02on 6

02on

yx

yx

rR

rR

Any points in two straight lines can be eigenvectors.

6for 1

26

6

12

2-4-

210

1-

2

22

25

1for2

1

42-

4-5

2

1

22

25

ex.

4) Diagonalizing a Matrix ( 행렬 대각화 )

6.for 622

625 1,for

22

25

222

222

111

111

yyx

xyx

yyx

xyx

.difference no :60

01 ,

10

06 cf.

DD

tors)(unit vec

5

1

5

25

2

5

1

where,

60

01

22-

2-5

21

21

21

21

C

CDMCyy

xx

yy

xx

Representing with the matrix operations,

.

,1

11

DMCC

DCDCMCCCDMC

DMCC 1

- Matrix D has elements different from zero only down the main diagonal,

diagonal matrix.

- D is called similar to M.

- When we obtain D given M, we have diagonalized M by a similarity

transformation.

5) Meaning of C and D (C 와 D 의 의미 )

(x’, y’) rotated through from (x, y)

.' ,

).','(' and ),( where'

.lyspecifical cossin

sincos where'

cos'sin'

sin'cos'

1 rDrMCCRrMCRCMrR

YXRYXRCRR

CCrr

yxy

yxx

- D=C-1MC is the matrix which describe in the (x’, y’) system the same

deformation (or transformation) that M describes in the (x, y) system.

- If C is chosen to make D=C-1MC diagonal, the new axes are along the

directions of the eigenvectors of M.

- The matrix C which diagonalizes M is the rotation matrix when the (x’, y’)

axes are along the directions of the eigenvectors of M.

12. Applications of diagonalization ( 대각화의 응용 )

- Central conic section (ellipse or hyperbola) with center at the origin

Ky

xMyxK

y

x

BH

HAyx

KByHxyAx

or

,2 22

.'

'''

'

'''

''''or cossin

sincos''

'

'

'

'

cossin

sincos

111

1

Ky

xDyx

Ky

xMCCyxK

y

xMCCCCyxK

y

xMyx

CyxCyxyxyxyx

y

xC

y

x

y

x

T

If C is the matrix which diagonalizes M, the above is the equation of the conic relative to its principal axes.

Example 1. 30245 22 yxyx

.arccos cf. .30'6''

'

60

01''

.60

01

section) previous (from .6 ,1 seigenvalue ,22

25

.3022

2530245

51matrixrotation

51

52

52

51

22

1

22

Cyxy

xyx

DMCC

M

y

xyxyxyx

This idea can be applied to three (or more) dimensions.

Example 2. 24226 222 zyzyxyx

.24

110

123

031

z

y

x

zyx

.3,4,1

)3)(4)(1(1213

10

230

10

133

11

121

0

110

123

031

matrix thisofequation sticCharacteri

3

.24'3'4'or 24

'

'

'

300

040

001

''' 222

zyx

z

y

x

zyx

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

3

110

123

031

,4

110

123

031

,1

110

123

031

equation, above the to3,4,1 Applying

110

123

031

Let’s find C.

. 0

.3 when ,, ;4 when ,, ;1 when ,0,

141

351

103

142-

355

143-

353-

101

141

142-

143-

351

355

353-

103

101

C

Example 3. Find the characteristic vibration frequencies

.21

12,

10

01

).222()(

,

21

21

22212

212

212

21

2221

y

xyxkV

y

xyxmT

xyyxkyxkkykxV

yxmT

3,1 ,0313421

12

s,eigenvalue thefind To

2

(continued)

2221

1

'3'

,30

01

21

12 ,

'

'

,in matrix thechange variable themake To

yxkV

CCy

xC

y

x

V

.''

. ,'

'

,in variablenew a find To

2221

11

yxmT

UCCUCCy

xC

y

x

T

Lagrange’s equation

m

k

m

k

tBytAx

kyymkxxm

y

L

y

L

dt

d

x

L

x

L

dt

d

yxkyxmVTL

3 ,

.conditions initial depending sin' ,sin'

'.3' ,''

0''

,0''

'3'''

21

21

222122

21

(continued)

Finding the orthogonal transformation matrix C,

.2

1 ,

2

1

2

1

2

12

1

2

1

3for 2

1

2

1 ,1for

2

1

2

1

3for 3 21

12 ,1for 1

21

12

yxyyxxy

xC

y

x

C

y

x

y

x

y

x

y

x

(continued)

phase ofout ).)(( .sin22

' .0' ,0For

phasein ).)(( .sin22

' .0' ,0For

,sin ,sin

2

1

21

tBy

yxxA

tAx

yxyB

tBytAx

‘Characteristic (or normal) modes of vibration’‘Characteristic (or normal) frequencies of the system’

chapter 3 linear algebra

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