Download Linear Algebra

N968200 高等工程數學
※ 先修課程：微積分﹑工程數學(一)-(三)
1.抽象代數導論(Introduction to Abstract Algebra)
2.張量分析(Tensor Analysis)
3.正交函數展開(Orthogonal Function Expansion)
4.格林函數(Green's Function)
5.變分法(Calculus of Variation)
6.攝動理論(Perturbation Theory)
Reference:
1.
Birkhoff, G., MacLane, S., A Survey of Modern Algebra, 2nd ed, The Macmillan Co, New York, 1975.
2.
徐誠浩, 抽象代數-方法導引, 復旦大學, 1989.
3.
Arangno, D. C., Schaum’s Outline of Theory and Problems of Abstract Algebra, McGraw-Hill Inc, 1999.
4.
Deskins, W. E., Abstract Algebra, The Macmillan Co, New York, 1964.
5.
O’Nan, M., Enderton, H., Linear Algebra, 3rd ed, Harcourt Brace Jovanovich Inc, 1990.
6.
Hoffman, K., Kunze, R., Linear Algebra, 2nd ed, The Southeast Book Co, New Jersey, 1971.
7.
McCoy, N. H., Fundamentals of Abstract Algebra, expanded version, Allyn & Bacon Inc, Boston, 1972.
8.
Hildebrand, F. B., Methods of Applied Mathematics, 2nd ed, Prentice-Hall Inc, New Jersey, 1972..
9.
Burton, D. M., An Introduction to Abstract Mathematical Systems, Addison-Wesley, Massachusetts, 1965.
10. Grossman, S. I., Derrick, W. R., Advanced Engineering Mathematics, Happer & Row, 1988.
11. Hilbert, D., Courant, R., Methods of Mathematical Physics, vol(1), 狀元出版社, 台北市, 民國六十二年.
12. Jeffrey, A., Advanced Engineering Mathematics, Harcourt, 2002.
13. Arfken, G. B., Weber, H. J., Mathematical Methods for Physicists, 5th ed, Harcourt, 2001.
14. Morse, F. B., Morse, F. H., Feshbach, H., Methods of Theoretical Physics, McGraw-Hill College, 1953
~ from Wikipedia
David Hilbert
The finiteness theorem
Axiomatization of geometry
The 23 Problems
Formalism
Born January 23, 1862 Wehlau, East Prussia
Died February 14, 1943 Göttingen, Germany
Residence
Germany
Nationality
German
Field Mathematician
Erdős Number
4
Institution University of Königsberg and Göttingen University
Alma Mater University of Königsberg
Doctoral Advisor Ferdinand von Lindemann
Doctoral Students Otto Blumenthal
Richard Courant
Max Dehn
Erich Hecke
Hellmuth Kneser
Robert König
Erhard Schmidt
Hugo Steinhaus
Emanuel Lasker
Hermann Weyl
Ernst Zermelo
Known for Hilbert's basis theorem
Hilbert's axioms
Hilbert's problems
Hilbert's program
Einstein-Hilbert action
Hilbert space
Societies Foreign member of the Royal Society
Spouse
Käthe Jerosch (1864-1945, m. 1892)
Children Franz Hilbert (1893-1969)
Handedness Right handed
Philip M. Morse
Founding ORSA President (1952)
B.S. Physics, 1926, Case Institute;
Ph.D. Physics, 1929, Princeton
University.
Faculty member at MIT, 1931-1969.
Methods of Operations Research
Queues, Inventories, and Maintenance
“Operations research is an
Library Effectiveness
Quantum Mechanics
applied science utilizing all known
Methods of Theoretical Physics
scientific techniques as tools in
Vibration and Sound
solving a specific problem.”
Theoretical Acoustics
Thermal Physics
Handbook of Mathematical Functions, with Formulas,
Graphs, and Mathematical Tables
Francis B. Hildebrand
George Arfken
Introduction to Abstract Algebra
抽象代數導論
• Preliminary notions
• Systems with a single operation
• Mathematical systems with two operations
• Matrix theory: an algebraic view
可交換性
a b  ba
抽象代數系統
反元素
R,, R,, R,,,V R
a  a 1  a 1  a  e
可交換群
單位元素
ae  ea  a
可交換單子
結合性
a  b  c  
a  b   c
可交換半群
向量
V , 
R, 
封閉性
群胚
V R 
R,,
=
(V ,), (R,,),
向量空間
純量
可交換環
可交換
單子環
域
半群
單子
群
R,
環
單子環
、  對  之分佈性
a  b  c   a  b   a  c 
b  c   a  b  a   c  a 
群胚 Groupoid
•
A goupoid R,  must satisfy
R is closed under the rule of combination
a, b  R  a  b  R
• Ex. Consider the operation  defined on the set S= {1,2,3}
by the operation table below.
From the table, we see
*
1
2
3
1
1
2
3
2
3
1
2
3
2
3
1
2 (1 3)=2 3=2 but (2  1) 3=3 3=1
The associative law fails to hold in this groupoid(S, )
半群Semigroup
• A semigroup is a groupoid whose
operation satisfies the associative law.
a, b  R  a  b  R (groupoid)
a, b, c  R  a  b  c  a  b  c
• Ex. If the operation  is defined on R# by a b = max{ a,
b },that is a  b is the larger of the elements a and b, or
either one if a=b.
a (b  c) = max{ a, b, c } = (a  b) c
that shows (R # , ) to be a semigroup
• If a,b, c, d  R and (R, ) is a semigroup, then
(a  b)  (c  d)  a  ((b  c)  d)
proof.
a  ((b  c)  d)  a  (b  (c  d))
 a  (b  x)
 (a  b)  x
 (a  b)  (c  d)
denoted (c  d) by x
單子 Monoid
•
A semigroup R,  having an identity element e
for the operation  is called a monoid.
a, b  R  a  b  R
(groupoid)
a, b, c  R  a  b  c  a  b  c (semigroup)
a  Re  R  a  e  e  a  a
Ex. Both the semigroups (SU , ) and
of monoids
A A A
(SU , )
are instances
for each A  U
The empty set is the identity element for the union
operation.
A U U  A  A
for each A  U
The universal set is the identity element for the
intersection operation.
群 Group
•
A monoid R,  which each element of R has
an inverse is called a group
(groupoid)
a, b  R  a  b  R
(semigroup)
a, b, c  R  a  b  c  a  b  c
(monoid)
a  Re  R  a  e  e  a  a
a  Ra 1  R  a  a -1  a -1  a  e
• If R,  is a group and a,b  R ,then (a  b)-1  b -1  a -1
Proof. all we need to show is that
(a  b)  (b -1  a -1 )  (b -1  a -1 )  (a  b)  e
from the uniqueness of the inverse of a  b
we would conclude (a  b)-1  b -1  a -1
(a  b)  (b -1  a -1 )  a  ((b  b -1 )  a -1 )
 a  (e  a -1 )
 a  a -1
e
a similar argument establishes that (b -1  a -1 )  (a  b)  e
Commutative 可交換性
a, b  R  a  b  b  a
a, b  R  a  b  Rgroupoid
Commutative
groupoid
a, b, c  R  a  (b  c)  (a  b)  csemigroup
Commutative
semigroup
a  Re  R  a  e  e  a  amonoid
Commutative monoid
a  Ra 1  R  a  a -1  a -1  a  egroup
Commutative group
•
Ex. consider the set of number S  {a  b 2 | a,b  Z}
and the operation  of ordinary multiplication, and Z
represents integer.
1. Closure:
a, b, c, d  Z  (a  b 2 )  (c  d 2 )  (ac  2bd)  (ad  bc) 2  S
2. Associate property a,b, c, d, e, f  Z

 

(a  b 2 )  (c  d 2 )  (e  f 2 )  (a  b 2 )  (c  d 2 )  (e  f 2 )
3. Identity element
1  10 2
4. Commutative property a,b,c, d  Z
(a  b 2 )  (c  d 2 )  (c  d 2 )  (a  b 2 )
 (S, )
is a commutative monoid.
Ring 環
•
A ring (R, , ) is a nonempty set R with two binary
operations  and  on R such that
1. (R, ) is a commutative group
a, b  R  a  b  b  a
a, b  R  a  b  Rgroupoid
a, b, c  R  a  (b  c)  (a  b)  csemigroup
a  Re  R  a  e  e  a  amonoid
a  Ra 1  R  a  a -1  a -1  a  egroup
2. (R, ) is a semigroup
a,b  R  a  b  Rgroupoid
a,b, c  R  a  (b  c)  (a  b)  csemigroup
3. The two operations are related by the distributive
laws a,b, c  R  a  (b  c)  (a  b)  (a  c)
(b  c)  a  (b  a)  (c  a)
A ring (R, , ) consists of a nonempty set R and two
operations, called addition and multiplication and denoted
by  and  , respectively, satisfying the requirements:
•
1.
2.
3.
4.
5.
6.
7.
8.
R is closed under addition
Commutative
Associative
Identity element 0
Inverse
R is closed under multiplication
Associate
Distributive law
a, b, c  R  1. a  b  R
2. a  b  b  a
3. a  (b  c)  (a  b)  c
4. 0  R  a  0  0  a  a
5. a -1  Rgroup  a  (-a)  0
6. a  b  R
7. a  (b  c)  (a  b)  c
8. a  (b  c)  (a  b)  (a  c)  (b  c)  a  (b  a)  (c  a)
Monoid Ring 單子環
• A monoid ring (R, , ) is a ring with identity that is a
semigroup with identity
a, b, c  R  a  b  b  a
 a  b  Rgroupoid
 a  (b  c)  (a  b)  csemigroup
e  R  a  e  e  a  a monoid
a 1  R  a  a -1  a -1  a  e group
Ring
 a  b  Rgroupoid
 a  (b  c)  (a  b)  c semigroup
 a  (b  c)  (a  b)  (a  c) 
(b  c)  a  (b  a)  (c  a)
 e  R  a  e  e  a  a monoid
Monoid ring
• Ring (R, , ) with commutative property
a,b,c  R  a  b  R 
ab  ba
a  (b  c)  (a  b)  c 
e  R  a  e  e  a  a 
a -1  R  a  a -1  a -1  a  e 
a b R 
a  (b  c)  (a  b)  c 
a  (b  c)  (a  b)  (a  c)  (b  c)  a  (b  a)  (c  a) 
a b  ba
a, b,c  R  a  b  b  a
 a bR
 a  (b  c)  (a  b)  c
e  R  a  e  e  a  a
a  1  R  a  a - 1  a - 1  a  e
 ab R
 a  (b  c)  (a  b)  c
 a  (b  c)  (a  b)  (a  c) 
(b  c)  a  (b  a)  (c  a)
 e  R  a  e  e  a  a
Commutative
Commutative monoid Ring
Subring 子環
•
The triple (S, , ) is a subring of the ring (R, , )
1. S is a nonempty subset of R
2. (S, ) is a subgroup of (R, )
3. S is closed under multiplication 
S  R
a,b,c  S  a  b  b  a 
a bS 
a  (b  c)  (a  b)  c 
e  S  a  e  e  a  a 
a -1  S  a  a -1  a -1  a  e 
a bS
•
The minimal set of conditions for determining subrings
Let (R, , ) be ring and   S  R Then the triple (S, , )
is a subring of (R, , ) if and only if
a,b  S  a - b  S
1. Closed under differences
2.
•
Closed under multiplication
a bS
Ex. Let S  {a  b 3 | a,b  Z} then (S, , ) is a subring
of (R # , , ), R# is a set of real numbers , since
a, b, c, d  Z, Z is the set of integers
(a  b 3 ) - (c  d 3 )  (a - c)  (b - d) 3  S
(a  b 3 )  (c  d 3 )  (ac  3bd)  (bc  ad) 3  S
This shows that S is closed under both differences and
products.
Field 域
• A field (F, , ) is a commutative monoid ring in
which each nonzero element has an inverse under
Definition of Field
A field is a mathematic al system ( F ,,) consisting of nonempty
set F and on F , called addition and multiplica tion, such that
(1) ( F , ) is a commutativ e group, with identity 0;
(2) ( F  {0},) is a commutativ e group, with identity 1;
(3) For each triple of elements a, b, c  F , a  (b  c)  a  b  a  c
Vector 向量
• An n-component, or n-dimensional, vector is an n
tuple of real numbers written either in a row or in a
column.
• Row vector  x1 , x2 ,, xn 
• Column vector  x1 
 x2 
 
 
 xn 
x k  R # called the components of the vector
n is the dimension of the vector
Vector space 向量空間
•
A vector space( or linear space) ((V,  ),(F,  ,  ),  )orV(F)
over the field F consists of the following:
1.
A commutative group (V, ) whose elements are called
vectors.
a,b  V  a  b  b  a
a,b  V  a  b  Vgroupoid
a,b,c  V  a  (b  c)  (a  b)  csemigroup
a  Ve  V  a  e  e  a  amonoid
a  Va 1  V  a  a -1  a -1  a  egroup
2.
A field (F, , ) whose elements are called scalars.
a, b,c  F  a  b  b  a
 a b F
 a  (b  c)  (a  b)  c
e  F  a  e  e  a  a
a 1  F  a  a -1  a -1  a  e
 a b F
 a  (b  c)  (a  b)  c
 a  (b  c)  (a  b)  (a  c) 
(b  c)  a  (b  a)  (c  a)
 e  F  a  e  e  a  a
 a 1  F  a  a -1  a -1  a  e
3.
An operation 。 of scalar multiplication connecting the group
and field which satisfies the properties
(a)  c  F and x  V , there is defined an element c x  V ;
V is closed under left multiplication by scalars
(b) (c 1  c 2 ) x  (c 1 x )  (c 2 x );
(c) (c 1 c 2 ) x  c 1 (c 2 x ),
(d) c ( x  y )  (c x )  (c y );
(e) 1 x  x, where 1 is the field identity element.
• Ex:
Let the commutativ e group be ( Mm  n,), where Mm  n is the set
of all m  n matrices and  is the operation of matrix addition.
For c  R # and (aij)  Mm  n, define scalar multiplica tion by
c(aij)  (caij)  Mm  n ← Vector Space
When m = n, we denote the particular vector space by Mn(R#)
Subspace 向量子空間
• Let V(F) be a vector space over the field F
W  V,W  
W(F) is a subspace of V(F)
The minimum conditions that W(F) must satisfy to be a subspace are:
(1) (W ,) is a subgroup of (V ,);
(2) W is closed under scalar multiplica tion.
x, y  W implies x  y  W ;
x  W and c  F imply cx  W .
• If V(F) and V’(F) are vector spaces over the same field, then the
mapping f : V → V’ is said to be operation-preserving if
f ( x  y )  f ( x)  f ( y ),
f (cx)  cf ( x),
f preserves
 pair of elements x, y V and c  F .
V(F) and V’(F) are algebraically equivalent whenever there exists a
one-to-one operation-preserving function from V onto V’
Linear Transformations 線性轉換
• Let V and W be vector spaces. A linear transformation from V into
W is a function T from the set V into W with the following two
properties:
(i) T ( x  y )  T ( x)  T ( y ), x, y  V .
(ii) T (x)  T ( x),  x  V and scalars  .
•x
T
•T(x)
W
V
T is function from V to W, {T ( x) | x  V }
Let V and W be vector spaces over the field F and let T be a
linear transformation from V into W.
• If V is finite-dimensional, the rank of T is the dimension of the range of T
and the nullity of T is the dimension of the null space of T.
• The null space (kernal) of T is the set of all vectors x in V such that T(x) = 0
ker T  { x V | T ( x )  0}
V
ker T
W
•
•
•
T
•
•0
ran T
The Algebra of Linear Transformations
•Let T : U → V and S : V → W be linear transformations, with U, V, and
W vector spaces.
( S  T )( x)  S (T ( x)), for x in U
The composition of S and V
if x1 and x2 are vectors in U, then
(S T )( x1  x2 )  S(T ( x1  x2 ))
(by definition of S T )
 S (T ( x1 )  T ( x2 ))
(by linearity of T )
 S(T ( x1 ))  S(T ( x2 ))
(by linearity of S )
 (S T )( x1 )  (S T )( x2 ) (by definition of S T )
Similarly, we have, with x in U and  a scalar,
(S T )( x )  S(T ( x )) (by definition of S T )
 S(T ( x ))
(by linearity of T )
  S(T ( x ))
(by linearity of S )
  (S T )( x ) (by definition of S T )
Representation of Linear Transformations by Matrices
線性轉換的矩陣表示
Let V be an n-dimensional vector space over the field F. T is a linear
transformation, and α1, α2,…,αn are ordered bases for V. If
T (1 )  a111  a21 2    an1 n
T ( 2 )  a121  a22 2    an 2 n

T ( n )  a1n1  a2 n 2    ann n
 T [1 ,  2 , ,  n ]  [T (1 ), T ( 2 ), , T ( n )]
 (1 ,  2 , ,  n ) A
 a11 a12  a1n 
a

a

a
21
22
2n 
其中 A  


 


a
a

a
n2
nn 
 n1
稱A為Linear Transformation T在
α1, α2,…,αn 下的矩陣
Inner Product 向量內積
Let a  a1i  a2 j  a3 k and b  b1i  b2 j  b3 k be two vectors in R 3
 the inner product of a and b, written (a, b ) which is denoted by a  b
a T b  [a1
a2
 b1 
a3 ]b2   a1b1  a2b2  a3b3
b3 
Certain properties of the inner product follow immediatel y from the
definition .
a, b, c are vectors in R 3 , and  and  are real scalars
(1) (a, a )  0
(a, a)  0 if and only if a  0
(2) (a, b )   (a, b )
(3) (a, b  c)  (a, b )  (a, c)
(a  b, c)  (a, c)  (b, c)
(4) (a, b )  (b, a)
It follows from the Pythagorean theorem that the length of the vector
a  a1i  a2 j  a3 k is a1  a2  a3
2
2
2
The length of the vector a is denoted by a
(a, a)  a1  a2  a3  a  (a, a)1/ 2
2
2
2
* Let a and b be nonzero vectors in R 3 and let  be the angle between them
 (a, b)  a b cos 
 b  a  a  b  2 a b cos 
2
2
z
2
a1  a2  a3
2
2
y
2
|b - a|
a
a2
b
a3
y
a1
a1  a2
2
x
2
θ
a
x
Inner Product Space 向量內積空間
If V is a real vector space, a function that assigns to every pair of vectors
x and y in V a real number ( x,y) is said to be an inner product on V , if it has
the following properties .
(1) ( x,x)  0
( x,x)  0 if and only if x  0
(2) ( x, y  z )  ( x, y )  ( x, z )
( x  y , z )  ( x, z )  ( y , z )
(3) (x, y )   ( x, y )
( x,  y )   ( x, y )
(4) ( x, y )  ( y, x)
The vector space V , together with its inner product, is said to constitute an
inner product space
 1 
 1 
x     and y      ( x,y)  11   2  2     n  n  x T y
 n 
  n 
Eigenvalues and Eigenvectors
特徵值與特徵向量
Let T:V  V be a linear operator on a vector space V .
(a) An eigenvalue of T is a scalar  for which T(v)  λv for some nonzero vector v in V
(b) If  is an eigenvalue of T, hen an eigenvector of T for the eigenvalue  is any
nonzero vector v for which T(v)  λv.
(c) If  is an eigenvalue of T , then the eigenspace of T for the eigenvalue  is the set
{v  V|T(v)  λv} consisting of all the eigenvedct ors of T for  , plus 0.
4  1
Ex. Let T be the linear operator on R defined by T ( x)  
x.

2 1 
 1  4  1 1
 1  4  1 1
1
1 




Since T      
 3   and T      
2 ,






1
 2
 1  2 1  1
  2   2 1   2
it follows that 3 and 2 are eigenvalue s of T with correspond ing
2
1
1 
eigenvecto rs   and   , respective ly. In this case, the linear operator
1
 2
T fixes the lines through the origin determined by scalar multiples of
the eigenvecto rs.
2I+4j
4
3
2
1
I+2j
I+j
1
T
3I+3j
2
1
1
2
3
Diagonalization 對角化
A square matrix is said to be a diagonal matrix if all of its entries are
zero except those on the main diagonal:
1 0
0 
2

 

0 0
 0
 0 
 

 n 
A linear operator T on a finite-dimensional vector space V is diagonalizable if
there is a basis vector for V each vector of which is an eigenvector of T.
Orthogonalization of Vector Sets 向量的正交化
It is desireable , as in the preceeding section, to form from a set of s linearly
independen t vectors u1 , u 2 ,, u s an orthogonal set of s linear combinatio ns
of the original vectors. We first select any one of the original vectors,
Let v 1  u1 , and divide it by its length  e1 
u1
l ( u1 )
Then we choose a second vector u 2 and v 2  u 2  ce1
The requiremen t that v 2 be orthogonal to e1 leads to the determinat ion
(e1 , v 2 )  (e1 , u 2 ) - c(e1 , e1 )  0 or c  (e1 , u 2 )
 v 2  u 2  (e1 , u 2 )e1
Let e 2 
u2
, v 3  u 3  c1e1  c2 e 2  v 3  u 3  (e1 , u 3 )e1  (e 2 , u 3 )e 2
l (u 2 )
A continuati on of this process finally determines the sth member of the
required set in the form
s 1
us
es 
where v s  u s   (e k , u s )e k
l (u s )
k 1
Gram-Schmit orthogonalization procedure
Quadratic Forms 二次形式
A homogeneou s expression of second degree of the form
A  a11 x1  a22 x2    ann xn  2a12 x1 x2  2a13 x1 x3    2an 1,n xn 1 xn is calles a quadratic form.
2
2
If we write yi 
2
1 A
, we obtain the equations :
2 xi
a21 x1  a22 x2    a2 n xn  y2

an1 x1  an 2 x2    ann xn  yn
aij x j  yi
Equivalent
a11x1  a12 x2    a1n xn  y1
Ax=y
aij  a ji
The set of equations can be written in the form : A x  y, A  [aij ] is a symmetric matrix
A  ( x, y )  x T A x
Canonical Form 標準形式
Let the vector x be expressed in terms of x' by the equation x  Q x'
 A  (Q x' )T A Q x'  x' T QT A Q x' or A  x' T A' x'
where the new matrix A' is defined by the equation A'  QT A Q
↑Diagonal matrix
If the eigenvalues and corresponding eigenvectors of the real symmetric matrix
A are known, a matrix Q having this property can be easily constructed
A1e1  1e1 ,, A n en  n en
eigenvector
eigenvalue
Let a matrix Q be constructed in such a way that the elements of the unit vectors
e1, e2,….,en are the elements of the successive columns of Q:
e11 
 en1 
e 
e 
e1   12   en   n 2 

  
 
 
e1n 
enn 
e11 e21
e
e22
12

Q
 

e1n e2 n




en1 
en 2 


enn 
 1e11 2 e21
 e
2 e22
1 12

AQ
 


1en1 2 en 2
 n en1 
1 0
0 
 n en 2 
2
or A Q  Q  
 
  


 n enn 
0 0
0
 0 
 

 n 

Since the vectors e1,....., en are linearly independen t, it follows that | Q |  0
Thus the inverse Q -1 exists, and by premultipl ying the equal members of
the equation above by Q -1 we obtain the result Q -1A Q  [i ij ]
 Q T  Q -1 or Q T Q  I  Orthogonal Matrix
 A   i x'i
2
A ei  i ei
Ex: Let T be the linear operator on R3 which is represented in
the standard ordered basis by the matrix A
 5  6  6
A   1 4
2 
 3  6  4
 eigenvalue 1  1, 2  2, 3  2
3
 2
 2
eigenvecor 1   1 ,  2  1 ,  3  0
 3 
0
1 
 3 2 2
1 0 0
Then we have Q   1 1 0  Q 1 AQ  0 2 0  D
↑
 3 0 1
0 0 2 ↑Diagonal matrix
Orthogonal matrix