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Transcript of 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.”
Chapter 3 Linear Algebra
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
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
Chapter 3 Linear Algebra
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
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
Chapter 3 Linear Algebra
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
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
Chapter 3 Linear Algebra
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
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’