* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Linear Algebra
Quadratic form wikipedia , lookup
Cross product wikipedia , lookup
System of linear equations wikipedia , lookup
Hilbert space wikipedia , lookup
Singular-value decomposition wikipedia , lookup
Tensor operator wikipedia , lookup
Jordan normal form wikipedia , lookup
Polynomial ring wikipedia , lookup
Cayley–Hamilton theorem wikipedia , lookup
Matrix multiplication wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Euclidean vector wikipedia , lookup
Invariant convex cone wikipedia , lookup
Exterior algebra wikipedia , lookup
Geometric algebra wikipedia , lookup
Eigenvalues and eigenvectors wikipedia , lookup
Commutative ring wikipedia , lookup
Homomorphism wikipedia , lookup
Covariance and contravariance of vectors wikipedia , lookup
Oscillator representation wikipedia , lookup
Vector space wikipedia , lookup
Matrix calculus wikipedia , lookup
Cartesian tensor wikipedia , lookup
Four-vector wikipedia , lookup
Linear algebra wikipedia , lookup
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 ba 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 ae ea 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 Re 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 Re R a e e a a a Ra 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 Re R a e e a amonoid Commutative monoid a Ra 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 10 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 Re R a e e a amonoid a Ra 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 ab ba 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 ba a, b,c R a b b a a bR a (b c) (a b) c e R a e e a a a 1 R a a - 1 a - 1 a e ab 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 bS a (b c) (a b) c e S a e e a a a -1 S a a -1 a -1 a e a bS • 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 bS 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 Ve V a e e a amonoid a Va 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 ) a111 a21 2 an1 n T ( 2 ) a121 a22 2 an 2 n T ( n ) a1n1 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) 11 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