Transcription

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3 i I II I II II IC IIC I II

4 ii

5 iii I

6 iv II

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9 3. () m, n mn a ij (i =,..., m, j =,..., n) m n m n a a a n a a a n A =.... a m a m a mn A a ij A (i, j) A (i, j) A = (a ij ) i m, j n, () n A = (a ij ) i,j, A = (a ij ) ) A = (a a a n

10 4 n m m A =. () n n n () n A = (a ij ) i,j A (k, k) a kk (k =,..., n) A.3 A, B A = B a a. a m () A, B m n () (i, j)..4 ( ) () A B A = (a ij ) B = (b ij ) m n A + B := (a ij + b ij ) () A B A B := (a ij b ij )

11 ( ) A = (a ij ) m n B = (b ij ) m n n = m A B AB ( m ) AB := a ik b kj k= m = n = m.6 ( ) A = (a ij ) m n λ λa λa := (λa ij )..7 A, B m n () ( ) A + B = B + A. () ( ) (A + B) + C = A + (B + C)..8 O O () O A A + O = O + A = A

12 6 () A A + X = X + A = O X () X A.9 A m n B n s C s t (AB)C = A(BC). n i, j =,..., n e ij {, i = j e ij =, i j n E n = (e ij ) n E n E. n δ ij {, i = j δ ij =, i j E n = (δ ij )

13 .. 7. m, n E m, E n m n m n A E m A = A, AE n = A..3 m,n λ, µ m n A, B () (λ + µ)a = λa + µa. () (λµ)a = λ(µa). (3) A = A. (4) λ(a + B) = λa + λb...4 A, B n AB, BA AB = BA.5 A, B n AB = BA A B AB BA A B.6 n A, B AB = O A O B O O n n.7 A B

14 8.8 A, B n E n n AB = BA = E n B A A B B A A B.9. n n A A. ( ) A n k A = E n, A = A, A = A A,..., A k = A k A. ( ) A, B n k, l () (.) A k A l = A k+l. (.) (A k ) l = A kl. () (AB) k = A k B k.3 ( ) A m n A = (a ij ) A t A t A := (a ji )..4 ( ) A n () A A = t A () A A = t A

15 ( ) A n () A m A m = O () A A = A.6 (, ) A n A = (a ij ) () A i > j a ij = () A i < j a ij =..7 ( ) A n A = (a ij ) A tr(a) tr(a) = n a ii. i=. ( ) A = () A () A n

16 () ( ) ( ) ( ) A = =. () ( ) ( ) ( ) A 3 = A 3 A = =, ( ) ( ) ( ) A 4 = A A = =. ( ) A n n = n (i) n = (ii) n = k n = k + ( ) ( ) ( ) A k+ = A k k k + A = =.. A 3 3 A (i, j) a ij a ij = i + j. A

17 .. A a = + =, a = + = 3, a 3 = + 3 = 4, a = + = 3, a = + = 4, a 3 = + 3 = 5, a 3 = 3 + = 4, a 3 = 3 + = 5, a 33 = = A = y = x + 5x y = x y 3 = x + 4x x = ( x x ), y = y y y 3 y = Ax A 5 3 A = 4

18 .4 ( ) A A = ( a A = c ) b d ( A a = c ) ( b a d c ) ( ) b a + bc ab + bd =. d ac + cd bc + d a + bc = c(a + d) = d + bc = b(a + d) = a + d c = a =, d = a + d. A = ( ), B = () A 3B + C ( ), C = ( ) () xa + yb + zc = O x, y, z x =, y =, z =

19 A =, B = 5 3 () 5X + A = B X () X + Y = A, X Y = B X, Y (3) X 5Y = A, 3X + Y = B X, Y ( ) ( ) A = 3 4, B =, C = 3, D = 5, E = () A () B (3) E.4 n n ( ) a b.5 () A = A (a+d)a+(ad bc)e = c d O ( ) 3 () A = A 3 3A 7A + 3E.6 (i, j) i + j 4 3

20 4.7 5 A = (a ij ) () a ij = ( ) i+j. i + j, i < j () a ij = i j, i > j, i = j ( ) ( ) ( ) A =, B =, C = () CA BA () (B 3C)A (C + B)A (3) (B + C)(B C)A.9 a, b, c x, y, z 3 x l l a 5 4 y = m, m = b. 3 4 z n 5 3 n c. { y = x x + 3x 3 () () y = x + 4x 3. ( ) 3 x () x y z 5 y 3 5 z y = 3x x y = x y 3 = x 3x (3) y = x 3 y = 3x y 3 = x

21 .. 5 ( ) x () x y z y z ( ) ( ) ( a. E =, I =, A = b () A t A t AA () A, t A E I ) b a.3 n A, B, C () λ(ab) = (λa)b = A(λB). () (AB)C = A(BC). (3) A(B + C) = AB + AC. (4) t ( t A) = A. (5) t (A + B) = t A + t B. (6) t (AB) = t B t A..4. ( ) 5 ( ) () 3 ()

22 6 ( a.5 A = c ) b d ( ) () B = AB = BA a, b, c ( ) () C = AC = CA a, b, c (3) ( ) ( ) A =, B = () AB BA. () (AB) = A B..7 n A, B () tr(a + B) = tr(a) + tr(b). () tr(λa) = λ tr(a). λ (3) tr(a) = tr( t A). (4) tr(ab) = tr(ba)..8 r AB BA = re n A, B

23 ( ) () () a. A = a () A, A 3 () A n. n A, X, Y AX = E n, Y A = E n X = Y. A n () A + t A () A t A (3) A 3 5 (4) A = A 5.3 () n X X = O

24 8 () n A A = B + C, A = B + C B, B C, C B = B, C = C.4 () x n. () A E A E (3) A (E A) A.5 A, B () AB AB = BA () AB = BA AB.6 A, B AB = BA. () A + B () AB.7 A () A () A 3. (3) A n

25 A = () A, A 3 () A n.9 A = A n ( ).3 α A A α =.3 A, B n () A, B AB () A, B AB.3 A, B [A, B] [A, B] := AB BA () [A, B] = [B, A]. () [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = O..33 A t A

26 .34 A A 4A + 3E = O () A () A A.35 A, B n AB = O A, B

27 . x, y, z x + y + 6z = 6 x + 3y + 4z = 9 3x + 5y + 3z = 5 () 3 3 x y z A x, y, z () E E =

28 . B = E A B 3 A A (3) E C 4 E =, C = 8 3 C = E 3 E B E 3 (4) D D = D = E 4 C E 4 (5) () (4) x, y, z. 6 6 x 6 () A = 3 4, 3 4 y = z () B = 3 4 = (3) E B = 8 =

29 E B 3 C E 3 = = 8 = C. 5 E 3 =. (4) C D E 4 = = = D. 4 E 4 = 8. 3

30 4 (5) D = E 4 E 3 E E A x x x y = D y = E 4 E 3 E E A y z z z 4 6 = =. 5 x =, y =, z = 5.. x x 3 = 4 x x + 3x 3 = 3x 8x + 6x 3 = x, x, x 3 =

31 .. 5 () () c (3) c () (3) ( ) ( ) ( ) ( 3) 4 ) 3 4 5

32 6 a a a n a a a n A = a m a m a mn (.) a a = a,..., a m, a i i (.) () a () a i = (i =,,..., m) () ( ) ( a a i ) ( i ) i (i, ) a ( a a i ) + a i = a i + a i = i =,..., m (), ()

33 .. 7 () () a a a n. B. B 3 B B. c j c j C =... ( c j c j c rjr ). O c rjr r + C r A C C r A r rank A = r A C C.

34 () () x x 3 = 4 x x + 3x 3 = 3x 8x + 6x 3 =

35 .. 9 x x x x, x, x 3 x x + 3x 3 = x 3x + 4x 3 = 3 3x 8x + 3x 3 =

36 3 x x x x x + 3x 3 = x + x x 3 = x + x + x 3 = 7 3 x, x, x 3 3. Ax = b x x + 3x 3 = x 3x + 4x 3 = 3 3x 8x + 3x 3 =

37 .. 3 x x x { x x 3 = 3 x x 3 = x, x, x 3 x 3 = α x = α 3, x = α, x 3 = α (α ) 3 x + x x 3 + x 4 + 3x 5 = x x + x 3 + x 4 + 6x 5 = 3x + x 4x 3 3x 4 9x 5 = 3

38 3 x x x 3 x 4 x x =, x x 3 =, x 4 + 3x 5 = x 3 (= α), x 5 (= β) x =, x = α, x 3 = α, x 4 = 3β, x 5 = β (α, β ) rank A = rank(a b) = 3. Ax = b n ranka =rank(a b) = r n r

39 A m n () E... O E = O.. E A A a a a a a a = a 3 a 3 a 3 a 3 a a 3 () E E =... O.... c. O.. i < j

40 34 E A A a a a a c a a = ca + a ca + a a 3 a 3 a 3 a 3 A c. i > j E (3) E 3... O E 3 = c. O.. E 3 A A 3 a a a a a a = a a. c a 3 a 3 ca 3 ca 3 A E, E, E 3 E, E, E 3 E = E, E c c E3 c c

41 , c, c (c ). 3 A = (E A) =

42 E, E, E 3 P 3 P (E A) = 3 3 P (E A) = (P P A) (3 3 6 ) 3 P = 3 3, P A = = E.

43 P E, E, 3 P A = E A P AP = E A P. n A.3 3 A = 4 6 E A

44 38 3 Q = QA = 3 8 A A QA A = Q QA A 3 Q A A 3.3 n A rank A = n.4 n n Ax = b A n

45 ranka <rank(a b) ranka =rank(a b) = r n r rank A = n A A A x = A b E A b A E x A E E A b x A x x = A b. Ax =, a x + a x + +a n x n = a n x + a n x + +a nn x n =

46 4 x = x = = x n = x 3x + x 3 = 3x + 4x x 3 = 7x 5x 3 = x x x x 3 = α x = 7 α, x = 5 7 α, x 3 = α α = x = x = x 3 = x = x = = x n = α

47 rank A < n A 3.4 A = A.6 ( ) () () 4 (3) (4) 3 (5) (6) a 7 a

48 4 a.8 a a.9 x + 3y + 4z = () 3x + 7y + z = 5x y + 4z = 3 () (3) (4) { x + 3y + 4z + w = x + y z + w = 3 x + 3y + 4z = 3x + 7y + z = 5x y + 4z = 3 x + y + 3z + 4w = x + 3y + 4z + 5w = 3x + 4y + 5z + 6w =. a, b, c, d (a, b, c, d) ax by az + bw = bx + ay bz aw = cx dy + cz dw = dx + cy + dz + cw =

49 x, y, z, w a, b y + 4z + w = x + y + 3z + w = x + y + 3z + w = b x y + aw =. x + 5y + 5z + 3w = 7 3x + 4y + z + w = 5 x + 4y + z + 3w = a a

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51 a a a a = a a a a, a a a 3 a a a 3 = a a a 33 + a a 3 a 3 + a 3 a a 3 a 3 a 3 a 33 a a 3 a 3 a a a 33 a 3 a a n (p, q,..., s) (,,..., n) 3.3 a a a n a a a n = ±a p a q a ns.... a n a n a nn (p, q,..., s)

52 () () (3) (4) (5) (6) (7) () (,, 4, 3) () (3,,, 4) (3) (3, 4,, ) (4) (, 3, 4, ) 3.3 A = (a ij ) 4 A () a 3 a a 34 a 4 () a a 4 a 3 a 43 (3) a 4 a 3 a 3 a a a a 3 a a 3 a a 3 = a a a 3 a 33 3 a 33. a a a n a a n a a n = a a n a a n a nn nn 3.5 a a a 3 a a () () a 33 a 33 a 44 a 44 a a a 3 (3) a 33 a 34 a 4 a 4 a 44

53 ka ka ()... () kann a n,... a,n a,n A a a a 3 a a a 3 a a a 3 = a a a 3. a 3 a 3 a 33 a 3 a 3 a A a a 3 a a a a 3 a a 3 a = a a a 3. a 3 a 33 a 3 a 3 a 3 a a + a a a 3 a a a 3 a a a 3 a + a a a 3 = a a a 3 + a a a 3. a 3 + a 3 a 3 a 33 a 3 a 3 a 33 a 3 a 3 a 33

54 A k k ka a a 3 a a a 3 ka a a 3 = k a a a 3. ka 3 a 3 a 33 a 3 a 3 a A a + ka a a 3 a a a 3 a + ka a a 3 = a a a 3. a 3 + ka 3 a 3 a 33 a 3 a 3 a a a a 3 b b a 3 =. c c a

55 ( 3 ) = ( 3 ) = ( ) = 5 5 = ( ) 3 = a a 3 a a 3 () a a a 3 = a a a 3 a 33 3 a 33. () a a 3 a a 3 a a 3 = a 3 a a 3 a 3 a 33 a () () (3) (4) 4 3 5

56 k k k l m () k () l (3) l m k kl l l m k + l l + m m + k / /4 /3 () () 3 (3) (4) / 5/6 /3 3 5 / / (5) (6) (7) (8) (9) () () () (3) a a a 3 3. A = a a a 3 a = a 3 a 3 a 33 a 3 = a 3 a 3 a 33 a a a 3, a = a a a 3, A = a a a 3 A

57 () a a a 3 () a 3 a a (3) a + a a a 3 (4) a + a a + a 3 a 3 + a a a a 3 3. A = a a a 3 A a 3 a 3 a 33 a a ka 3 ka ka 3 ka () a a ka 3 () ka ka 3 ka a 3 a 3 ka 33 ka 3 ka 33 ka A n ka = k n A n A = (a ij ) i j (n ) D ij A (n ) ( ) i+j A (i, j) A ij A ij = ( ) i+j D ij. 3. ( ) n A = (a ij ) A = a j A j + a j A j + + a nj A nj = a i A i + a i A i + + a in A in. j i

58 A = a A + a A + a 3 A 3 = 3( ) + + ( )+ + 5( )+3 = 3( ) = () 3 () 3 (3) 3 (4) (5) (6) (7) (8) (9) ()

59 a a... = ( ) (n )n/ a a a n a n A = (a ij ) A = A A A = A A A 3 A A A 3 A A 3 A 3 A 33 A ij A (i, j) A =

60 54 3 A = A A = ( ) + = A =, A 3 = 3, A =, A =, A 3 = 6, A 3 = 5, A 3 = 7, A 33 = 3. A = = = ( ) ( ) ( ) ( ) 3 () () (3) (4) 3 4 (5) (6) (7) 3

61 ( ) a x + a x + a 3 x 3 = b a x + a x + a 3 x 3 = b a 3 x + a 3 x + a 33 x 3 = b 3 A = (a ij ) A = x = b a a 3 b A a a 3, x = a b a 3 a A b a 3, x 3 = a a b a A a b b 3 a 3 a 33 a 3 b 3 a 33 a 3 a 3 b x + x + x 3 = 3x + x + x 3 = x x + x 3 = 3 A A = 3 = 3

62 56 3 x = = 4, 3 3 x = 3 = 3 3, 3 x 3 = 3 = x = 4, x = 3, x 3 = 3. ( 4 + ) =, ( 3 ( 4) + ) =, ( ( 4) ) = x + x + x 3 = x x + x 3 = () x x + x 3 = () x + x x 3 = 3x + x 3 = x + 3x x 3 = x + x x 3 = x x + x 3 = (3) x x 3x 3 = (4) x x x 3 = x + x x 3 = x + x x 3 =

63 n A, B AB = A B. 3.5 n A A A = A = A 3.8 ( a c ) ( b x d z ) ( y ax + bz = w cx + dz ) ay + bw cy + dw (ad bc)(xw yz) = (ax + bz)(cy + dw) (ay + bw)(cx + dz) (a b )(x y ) = (ax + by) (ay + bx) 3.9 n A n P xe A = xe P AP. 3. A, B n B xe AB = xe BA. B

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65 ( ) A, B AB 3 AB = CD () AB = CD. AB = AB AB () AB // CD. AB, CD (3) AB CD a AB CD a a = AB 4. ( ) a b AB BC a b a + b a + b(= AB + BC) = AC

66 ( ) a a = AA, a(= AB) = BA a b a b = a + ( b) 4.4 ( ) a = AB k a k ka = k AB k ka ka, k > ka =, k = ( k)( a), k < k a 4.5 ( ) 8 () (a + b) + c = a + (b + c). () a a + = a (3) a a + b = b(= a) (4) a + b = b + a.

67 4.. 6 (5) (kl)a = k(la). (6) a = a. (7) (k + l)a = ka + la. (8) k(a + b) = ka + kb. 4.6 ( ) θ θ = AOB a = OA, b = OB a, b a b a b = a b cos θ 4.7 a = a a, cos θ = a b a b. a b a b =. a b () (a + b) c = a c + b c. () ka b = k(a b). (3) a b = b a. 4.9 a = OA, b = OB 3 O, A, B a = OA, b = OB, c = OC 3 O, A, B, C

68 a, b x a, b x = xa + yb x, y a, b, c x a, b, c x = xa + yb + zc x, y, y ( ) e, f 4. x = xe + yf x = (x, y) x 3 e, f, g 4. x = xe + yf + zg x = (x, y, z) x

69 ( ) P l P P = t l (t R) P (x, y ), P(x, y), l = (l, m) { x = x + tl y = y + tm lm t x x l = y y m P (x, y, z ), P(x, y, z), l = (l, m, n) x = x + tl y = y + tm z = z + tn lmn t x x l = y y m = z z n 4.3 t l (l, m), (l, m, n) l + m =, l + m + n =

70 l ( 4.7 ) n l π π n π 4.5 ( ) P l n l P n P P = l P (x, y ), P(x, y), n = (a, b) xy ax + by (ax + bx ) = P π n π P n P P = π P (x, y, z ), P(x, y, z), n = (a, b, c) xyz ax + by + cz (ax + bx + cz ) = ( ) a b a b a b a b a b

71 () a b = b a. () ka b = a kb = k(a b). (3) a (b + c) = a b + a c. 4.8 e, f, g e, f, g () e e = f f = g g =. () e f = f e = g. (3) f g = g f = e. (4) g e = e g = f. 4. p = (a, b), q = (x, y) p q (a, b) (x, y) = ax + by p = (a, b, c), q = (x, y, z) p q (a, b, c) (x, y, z) = ax + by + cz

72 66 4 e, f 4.7 = e = e e, = f = f f e f = e f cos 9 =. p = (a, b), q = (x, y) p = ae + bf, q = xe + yf. (a, b) (x, y) = p q = (ae + bf) (xe + yf) = axe e + aye f + bxf e + byf f = ax + by. e, f, g 4.7 = e = e e, = f = f f, = g = g g e f = e f cos 9 =, f g = f g cos 9 =, g e = g e cos 9 =. p = (a, b, c), q = (x, y, z) p = ae + bf + cg, q = xe + yf + zg. (a, b, c) (x, y, z) = p q = (ae + bf + cg) (xe + yf + zg) = axe e + aye f + aze g +bxf e + byf f + bzf g +cxg e + cyg f + czg g = ax + by + cz.

73 p = (a, b), q = (x, y) θ cos θ = ax + by a + b x + y 4. p = (a, b, c), q = (x, y, z) θ ax + by + cz cos θ = a + b + c x + y + z 4.3 p = (a, b), q = (x, y) q p a, b, x, y 4.4 p = (a, b, c), q = (x, y, z) q p a, b, c, x, y, z 4.5 p = (a, b), q = (x, y) a, b, x, y 4.6 p = (a, b, c), q = (x, y, z) a, b, c, x, y, z 4.7 p = (, ), q = ( 3, 3) p q 4.8 p = (,, ), q = (, +, ) p q

74 p = (a, b), q = (x, y) S a b S = x y y R Q q S p P O x OPRQ S S = OP OQ sin POQ. 4. cos POQ = ax + by a + b x + y sin POQ = cos POQ = (ax + by) (a + b )(x + y ) = (a + b )(x + y ) (ax + by) (a + b )(x + y ) = a y abxy + b x (ay bx) = (a + b )(x + y ) (a + b )(x + y ).

75 sin POQ (ay bx) = ay bx 4.3 sin POQ = ay bx a + b x + y. OP = p = a + b, OQ = q = x + y. S = a + b x + y a = ay bx = x b y. ay bx a + b x + y 4.9 P (x, y ), P (x, y ), P 3 (x 3, y 3 ) 4 x y x y x 3 y A(, 3), B(, ), C(3, ) ABC 4. 3 P (x, y ), P (x, y ), P 3 (x 3, y 3 ) x y x y = x 3 y 3

76 [ ] ax + by + c = c ab m = a a + b, n = b a + b, d = c a + b mx + ny = d () n = (m, n) () n =. (3) O (dm, dn). (4) d O () ab x b = y + c b a (b, a) n (b, a) = (m, n) (b, a) = bm an = a b a + b a b ba ab = a + b a + b = n

77 () n = n n = (m, n) (m, n) = a m + n = a + b + b a + b a + b = = a + b (3) () (m, n) (m, n) y n = x mx + ny = d m y = n x mx + ny = d m mx + n m x = d, (m + n )x = md. () m + n = x = md y = n m x x = md, y = nd (md, nd) (4) O O (3) () m +n = (3) (md) + (nd) = m d + n d = (m + n )d = d = d = d d

78 x 4y + 5 =, 5x + y 6 = () θ ( θ π ) cos θ () (3) O x 4y + =, 3x 4y 3 = P (x, y ), P (x, y ) x y x y = x y

79 () x l = y m, cos θ x l = y m θ () a x + b y + c =, a x + b y + c = θ cos θ (3) x = y = z 5, x 3 = y = z cos θ θ (4) x + 3y + z 5 =, x + y + z 3 = θ cos θ () x = y x, = y θ l m l m (l, m ), (l, m ) 4.7 cos θ = (l, m ) (l, m ) (l, m ) (l, m ) = l l + m m l + m l + m. () a x + b y + c =, a x + b y + c = θ (a, b ), (a, b ) 4.7 cos θ = (a, b ) (a, b ) (a, b ) (a, b ) = a a + b b a + b a + b.

80 74 4 (3) x = y = z 5, x 3 = y = z θ (,, 5) (3,, ) cos θ = = (,, 5) (3,, ) (,, 5) (3,, ) = = = (4) x + 3y + z 5 =, x + y + z 3 = θ (, 3, ), (,, ) cos θ = = (, 3, ) (,, ) (, 3, ) (,, ) = = 6 4 = P (x, y, z ), P (x, y, z ) 4.6 l = (l, m), l e, f θ, η l = cos θ, m = cos η

81 a = (a, a, a 3 ), b = (b, b, b 3 ) ( ) a a 3 a b = b b 3, a 3 a b 3 b, a a. b b a, b a = a e + a f + a 3 g, b = b e + b f + b 3 g 4.8 a b = (a e + a f + a 3 g) (b e + b f + b 3 g) = a b e e + a b e f + a b 3 e g +a b f e + a b f f + a b 3 f g +a 3 b g e + a 3 b g f + a 3 b 3 g g = a b g a b 3 f a b g + a b 3 e + a 3 b f a 3 b e = (a b 3 a 3 b )e + (a 3 b a b 3 )f + (a b a b )g a a 3 = b b 3 e + a 3 a b 3 b f + b b g a = (, 3, ), b = (,, 3) a b P (x, y, z ), P (x, y, z ), P 3 (x + x, y + y, z + z ) S = y z + z x x y + y z z x x y

82 O 3 P (x, y, z ), P (x, y, z ), P 3 (x 3, y 3, z 3 ) OP i 6 V x y z V = x y z x 3 y 3 z O 3 P (x, y, z ), P (x, y, z ), P 3 (x 3, y 3, z 3 ) x y z V = x y z = x 3 y 3 z a = (, 3, ), b = (,, 3), c = (,, ) 6

83 A n n P P AP A A 5. A n x α Ax = αx α A x α 5.3 A n α A x α 5.4 (αe n A)x = αe n A = A n xe n A = x n A

84 P n t P = P P 5.6 () n A ( O) n () A ( O) 5.7 A n () A () A n A = x 5 4 xe A = 3 x 4 4 = x 3 x 5x 3 6 x + 5 = (x + ) (x 3). A, 3 x y z

85 x y z = x = y, z = y x y = z x y z x y z. = y. x y z. x y z x x = y, z = y. y = y. z A, 3 k, k k, k 5. 3 A =

86 8 5 A x + 3 xe A = 3 x x + = x 3 x 4x 36 = (x + )(x + 3)(x 6), 3, 6 k, 3 k, 6 k 3 / / 3 /, / / 6 3 /, / 6 3 / 6 / / 3 / 6 P = / 3 / 6 / / 3 / 6 P AP = P 3 6

87 5.. 8 P AP = n ( ) A = 3 A x + xe A = x 3 = x x ± 3 + ( ) 3 k +, 3 3 ( ) k 3 ( ) P = AP = P ( + ) 3 3

88 8 5 ( ( A n + ) ) n 3 = P P 3 ( ( + ) 3) n = P ( 3) n P 5.4 n A A = E A λ A x λ Ax = λx A x = A(Ax) = A(λx) = λ(ax) = λ x A = E A x = Ex = x, λ x = x, (λ )x =. x λ = λ =, 5.5 {x n }, {y n } { x n = 4x n + y n y n = 3x n 7y n (x, y ) = (3, )

89 ( ) 4 A = 3 7 ( ) x n y n = A ( = A n ( x n y n ( ) A,, ( ) ( ) ( ) 5 5. P = P AP = 3 3 A n = ( ( ) ) n P P ( ) = ( ) n P P ( ) n ( ) = 6( ) n 5( ) n ( ) n ( ) n 3( ) n + 3( ) n 5( ) n + 6( ) n. (x, y ) = (3, ) { x y ) ) x n = 8( ) n 5( ) n y n = 4( ) n + 5( ) n. 5.6 x n+ 4x n+ 5x n = x =, x = {x n }

90 84 5 ( y n ) = x n+ y n+ = 5x n + 4y n A = 5 4 ( ) ( ) x n x n = A y n y n ( ) = A n x 5.5 x n = (( ) n +5 n )/3 y 5.7 { dx dt = x + x dx dt = 5x 3x (x (), x ()) = (, ) n A a,..., a n x. x n = P dx dt. dx n dt x = A. x n e a t e a t x () P. e ant x n ()

91 a P a AP = ( 5 3 a n ) ( dx dt dx dt A = = A ( ) ( A, 4, ( ) ( ) P = A = P 5 4 (x (), x ()) = (, ) { x = e t x = e t ) ( x x ) 5. ) P. 5.8 d y dt 3dy dt y = y() =, y () = z = dy dt dz dt ( ) = 3z + y A = 3 ( ) ( ) dy y dt dz = A z dt 5.7 y = ( e t +e 5t )/7

92 () 5 6 () 3 (3) a (4) 7 3 (5) a (6) a (7) 3 (8) () () a a () 3 () a (3) a b b 5.4 ( ) ( ) 3 7 () () () ()

93 ( ) ( n ) () () a b b a a b b a A 5.9 A A 3 = A 5. n A n B = A n B 5. A n Φ A (x) = xe A A () Φ A (x) x n tr(a) () Φ A (x) ( ) n A 5. A n P n Φ A (x) = xe A A () Φ P AP (x) = Φ A (x) () Φt A(x) = Φ A (x)

94 A ( n Φ ) A (x) = xe A A A A = (A, A n ) Φ A (x) = O A Φ A (x)φ A (x) 5.4 { () () {x n }, {y n } x n = x n 6y n y n = x n 3y n (x, y ) = (, ) x n = x n + y n + z n y n = x n + 3y n + z n z n = x n + y n + z n (x, y, z ) = (,, ) 5.5 x n+3 x n+ 4x n+ +4x n = x =, x =, x = {x n } 5.6 { () () x = 5x 8x (x x (), x ()) = (, ) = 3x 6x x = x + x x 3 x = 3x 5x + 3x 3 x 3 = 3x x 3 (x (), x (), x 3 ()) = (,, ) 5.7 y y 4y + 4y = y() =, y () =, y () =

95 II

96

97 K R C 6. V a, b V a + b V a V k K ka V V K a, b, c V k, l K (V) (a + b) + c = a + (b + c). (V) a + b = b + a. (V3) V a V a + = a. (V4) a a a + ( a) =.

98 9 6 (V5) k(la) = (kl)a. (V6) a = a. (V7) k(a + b) = ka + kb. (V8) (k + l)a = ka + la. V V a a 6. x x n K n =. x n x i K x y x + y x. + y. = x + y., x kx k x. = kx.. x n y n x n + y n x x x n kx n x =., y = y., z = z. Kn y z x n y n z n

99 (V3) (V4) =. x + = x x K n (V3) x x x x = (V4). x n (V), (V), (V5), (V6), (V7), (V8) (V) x + y z x + y + z x + y (x + y) + z =. + z. = x + y + z. x n + y n z n x n + y n + z n x y + z x =. + y + z = x + (y + z).. x n y n + z n (V) x + y y + x x + y x + y =. = y + x. = y + x. x n + y n y n + x n

100 94 6 (V5) (V6) lx k(lx ) (kl)x lx k(lx) = k. = k(lx ). = (kl)x. = (kl)x. lx n k(lx n ) (kl)x n x x x x x =. =. = x. x n x n (V7) x + y k(x + y ) x + y k(x + y) = k. = k(x + y ). x n + y n k(x n + y n ) kx + ky kx ky kx + ky =. = kx. + ky = kx + ky.. kx n + ky n kx n ky n

101 (V8) (k + l)x kx + lx (k + l)x (k + l)x =. = kx + lx. (k + l)x n kx n + ly n kx lx kx =. + lx = kx + lx.. kx n lx n 6. K n m Mat(n, m) 6. (a n ) n= A (a n ) + (b n ) = (a n + b n ), k(a n ) = (ka n ). A 6.3 [a, b] C(a, b) (f + g)(x) = f(x) + g(x), (kf)(x) = kf(x). f, g C(a, b) k R C(a, b) 6.4 K n x

102 n m A { x K m Ax = } 6. () V v, v,..., v n k v + k v + + k n v n = k = k = = k n = k i K ( i n) () V v, v,..., v n V x V x = k v + k v + + k n v n k, k,..., k n K v, v,..., v n V (3) V v, v,..., v n V 6.3 v, v,..., v n u, u,..., u m V n = m 6.4 V V dim V V 6. R n e, e,..., e n R n e i i..

103 k e + +k n e n = k k. =. k n k = k = = k n = e, e,..., e n x = x. x n x = x e + x e + + x n e n e, e,..., e n V e, e,..., e n V 6.6 n m Mat(n, m) nm 6.7 A () 4 () (3) (4)

104 R C(R) () {e x, e x } () {sin x, cos x, } (3) {e x, xe x } 6.9 k f k f k (x) = { sin πx, k x k +, n {f k } n k= C(R) 6. V v V 6. V {v,..., v n } w k := v + v + + v k {w, w,..., w n } V V U U V () x U, y U = x + y U. () x U, k K = kx U. 6.6

105 V x, x,..., x n { k x + k x + + k n x n k i K } V x, x,..., x n 6.8 U V dim U dim V dim U = dim V U = V 6.3 Mat(, ) {A = (a ij ) Mat(, ) a = a = a = a } V = {A = (a ij ) Mat(, ) a = a ( = ) a = a ( } ) A =, B = ( ) A, B V A + B = A + B V 6. Mat(, ) Mat(, ) Z E () {A = (a ij ) Mat(, ) a ij Z ( i, j )}

106 6 () {A Mat(, ) A = t A} (3) {A Mat(, ) det A = } (4) B Mat(, ) {A Mat(, ) AB BA = } (5) {A Mat(, ) t AA = E } 6.3 n A = (a ij ) A Tr(A) Tr(A) = i a ii { A Mat(n, n) Tr(A) = } Mat(n, n) 6.4 n B { A Mat(n, n) Tr(AB) = } Mat(n, n) B = 6.5 (a n ) n= n= a n < + 6. A n n 6.6 [a, b] D(a, b) 6.3 C(a, b) 6.7 n f n f (n) (x) = ( x R) 6.6 D(R) C(R)

107 V U, U U U V, V U + U = { u + u u U, u U } 6. V U, U V U + U U U 6. V U, U dim(u + U ) = dim U + dim U dim(u U ). 6.4 K 4 x =, x =, x =, x 4 = 3 3 x, x K 4 V, x 3, x 4 K 4 V V V V + V x V V x = k x + k x = k 3 x 3 + k 4 x 4 k + k = k 4 k + k = k 3 k + k = k 4 3k = 3k 3

108 6 k = t, k = 3t, k 3 = t, k 4 = 4t (t K) 4 V V x + 3x = 4 3 dim(v V ) = x, x x 3, x 4 dim V = dim V = 6. dim(v + V ) = dim V + dim V dim(v V ) = 3 x, x, x 3 V + V 6.8 K 4 V, V a b V = a + 3b + c d = c d a b a + b d = V = c 3a + 5b c d = d V, V K 4 K 4 V V V, V V + V = K 4 K 4 V, V V V

109 6.. 3 V + V a b V = a + b + c + 3d =, c d a b V = a 3b + c + d =. c d 6. x =, x = 3, x 3 = 5, x 9 4 = x, x K 4 V x 3, x 4 K 4 V V V V + V 6. x = 3, x 3 = 5, x 3 =, x 3 4 = 4, x 5 = x, x, x 3 K 4 V x 4, x 5 K 4 V V V V + V 6.3 V W, W V v W, v W W W = {}

110 V U, U V = U + U U U = {} V U U V = U U 6.3 V U, U x V x = x + x (x i U i ) 6.4 V U,..., U n (n ) x V x = x + + x n (x i U i ) V U,..., U n V = U U n n = V V, V, V 3 V + V = V V V = (V + V ) V 3 V = V V V 3 x V V x x = x + x + x 3 (x i V i ) x = x, x = x 3 = x = x, x = x 3 = x = V V = {} V + V = V V x (V + V ) V 3 x x = v + v (v i V i ) x = x + x + x 3 (x i V i ) x = v, x = v, x 3 = x = x =, x 3 = x (V + V ) V 3 = {}

111 V = V V V 3 V = (V + V ) V 3 V + V = V V V = (V + V ) V 3 x V x = x + x + x 3 (x i V i ) x = y + y + y 3 (y i V i ) x + x, y + y V + V V = (V + V ) V x + x = y + y x 3 = y 3 V + V = V V 6.3 x = y x = y V = V V V V W, W V = W + W V W W v W, v W v + v = v = v = 6.5 K n V V A n W := { Av v V }, W := { Av v V } W, W K n W W K U V f : U V () f(x + x ) = f(x ) + f(x ) (x, x U). () f(kx) = kf(x) (k K, x U).

112 C(R) R φ: C(R) R φ(f) = f() f, g C(R), c R φ(f + g) = (f + g)() = f() + g() = φ(f) + φ(g) φ φ(cf) = (cf)() = c(f()) = cφ(f) 6.6 f Mat(n, m) n m x x x () f. = x. f : Rn R n x n x n () f(x) = X + t X f : Mat(n, n) Mat(n, n). (3) A, B Mat(n, n) f(x) = AX XB f : Mat(n, n) Mat(n, n). (4) a K w K n f(v) = av + w f : K n K n

113 φ: V W () V = K 3, W = K φ φ x x = x + x x 3 () V, W K x φ(p(x)) p(x) n p (n) (x) (3) V, W K x 4 φ φ(p(x)) = p(x) + p (x) (4) V, W n Mat(n, n) φ P φ(a) = P AP 6.6 f : U V f(u) = { f(x) x U } V f Im f f () = { x f(x) = } U f Ker f 6.7 f : U V U dim U = dim(ker f) + dim(im f).

114 f : U V f : U V U V U = V 6.7 ( ) φ: K 3 K 3 A = φ(v) = Av Ker φ x v = y Av = x + y + 3z =, z y + z = Ker φ = t t K φ, Ker φ 6.9 f : V W Ker f = {} 6.3 V, W, f : V W Ker f = {} v,..., v n V f(v ),..., f(v n )

115 f : V W V v,..., v n f(v ),..., f(v n ) W f 6.9 f : U V U u, u,..., u m V v, v,..., v n a ij K ( i n, j m) f(u j ) = a j v + a j v + + a nj v n v, v,..., v n V a ij n m (a ij ) U u, u,..., u m V v, v,..., v n f 6. f : U V, g : V W α = {u, u,..., u m } β = {v, v,..., v n }, γ = {w, w,..., w p } U, V, W α, β f A, β, γ g B g f α, γ BA 6. V v, v,..., v n v, v,..., v n n A = (a ij ) v j = a j v + a j v + + a nj v n A v, v,..., v n v, v,..., v n 6. V v, v,..., v n v, v,..., v n V : V V v, v,..., v n v, v,..., v n

116 6 6.8 f : K n K m m n A f(x) = Ax e i K n K m i i () v,..., v n K n v,..., v n e,..., e n i v i (v... v n ) () K n e,... e n K m e,... e m f A (3) K n v,..., v n K m w,..., w m f (w... w m ) A (v... v n ) () v j = v j. v nj (v ij ) = (v v j = v j e + + v nj e n... v n ) () A = (a ij ) j a j a j = f(e j ) = (a... a n ) e j = a j = m a i,j e i f e,... e n e,... e m A (3) f v,..., v n w,..., w m X v,..., v n e,..., e n K n : K n K n B, e,... e n e,... e m i= a j. a mj

117 6.4. f A, e,..., e m w,..., w m K m : K m K m C X = CAB () B = (v... v n ), C = (w... w m ) (w... w m ) A (v... v n ) 6.3 V, W f : V W () V = K 3 v =, v =, v 3 = ( ) ( ) W = K w =, w = f A = ( ) 4 f(x) = Ax 3 () V = W = K 3,, f 3 A = f(x) = Ax 4 (3) V, W 3 x V W, x, x, x 3 f f(p(x)) = e x (e x p(x))

118 6 ( ) ( ) (4) V = W = Mat(, ) V W,, ( ) ( ) ( ), f P = f(x) = P X 3 4

119 K = R C V K V x, y (x, y) K V () (x + x, y) = (x, y) + (x, y). () (ax, y) = a(x, y) (a K). (3) (x, y) = (y, x) (c c ) (4) (x, x), (x, x) = x = K = R (3) (x, y) = (y, x) K V V K K = R V K = C V (x, x) ( ) (x, x) x x x = (x, x). V, (x, y) = x y x y x = x S S

120 4 7 V K (, ) () (.) (x, y + y ) = (x, y ) + (x, y ). (.) (x, ay) = a(x, y). (.3) x V, (x, y) = (x, z), y = z. () ( n i= a ix i, m j= b jy j ) = n i= m j= a ib j (x i, y j ). (3) (3.) ax = a x. (3.) x. x = x =. (3.3) (x, y) x y ( ) x y. (3.4) x + y x + y ( ) x = y = ax (a R, a ) (4) V x,, x r (5) - a,, a r (r ) V e = a a e = a a, a = a (a, e )e e 3 = a 3 a 3, a 3 = a 3 (a 3, e )e (a 3, e )e. e r = a r a r, a r = a r (a r, e )e (a r, e )e (a r, e r )e r e,, e r

121 7.. 5 (6) V V V (7) dim V = n e,, e r (r n) V (8) V x, y, cos θ = (x, y) x y ( ) (x, y) x y ( θ π) θ x y (x, y) x y x = y = (x, y) = π. < > 7. K n x = x., y = y. x n y n (x, y) = x y + + x n y n (= t x y) K n K n K n

122 6 7 < > K = R x = x., x = x., y = y. x n x n y n (x + x, y) = t (x + x ) y = (x + x )y + + (x n + x n)y n = (x y + + x n y n ) + (x y + + x ny n ) = t x y + t x y = (x, y) + (x, y). (ax, y) = t (ax y = (ax )y + + (ax n )y n = a(x y + + x n y n ) = a t x y = a(x, y). (x, y) = x y + + x n y n = x y + + x n y n = (y x + + y n x n ) = t y x = (y, x ). (x, x ) = x x + + x n x n = x + x n. (x, x ) = x = = x n = x = = x n = x =. (, ) K n 7. R n [x] n R R n [x] F, G (F, G) = F (t)g(t) dt R n [x]

123 7.. 7 R n [x] F, F, G, R a (F + F, G) = = (af, G) = (F + F )(t)g(t) dt = F (t)g(t) dt + = (F, G) + (F, G). = a (F, G) = (F, F ) = (af )(t)g(t) dt = F (t)g(t) dt F (t)g(t) dt = a(f, G). F (t)g(t) dt = F (t) dt = [, ] F (t) = F =. (, ) R n [x] (F (t) + F (t))g(t) dt af (t)g(t) dt G(t)F (t) dt = (G, F ). 7.3 (x, y) x y y = y a K x ay = (x ay, x ay) = (x, x ay) a(y, x ay) = (x, x) a(x, y) a(y, x) + aa(y, y).

124 8 7, a = (x, y)/(y, y), (x, x) (x, y) (y, y) = x (x, y) y (x, y) x y. x ay = x ay = x, y 7.4 R 3,, a =, a =, a 3 = a = + = e = a a = a = a (a, e )e = =., a = = 6 e = a a = 6 = 6.

125 7.. 9 a 3 = a 3 (a 3, e )e (a 3, e )e = ( + ) = =. 3 a 3 = = 3, e 3 = a 3 a 3 = 3 3,. =. 3 6, V x + y = x + y (x, y) =.

126 7 x + y = (x + y, x + y) = (x, x) + (x, y) + (y, x) + (y, y) = x + y + (x, y). ( R, (y, x) = (x, y) = (x, y).) x + y = x + y (x, y) =. x + y = x + y + (x, y) + (x, y) (x, y) + (x, y) = = (x, y) = ( ) ( ) x =, y = C (x, y) = + i + i i( + i) = i (x, y) + (x, y) = i + ( i) =. ( ) 7. A = R (x, y) = t xay (, ) R 7. V = M n (R) (A, B) = tr(b t A) (, ) V 7.3 V, V K V V {x, x }, {y, y } V V. ({x, x }, {y, y }) = (x, y ) + (x, y ). (, ) V, V ( ) V V (x, x ) {x, x }

127 (( ) ( )) () R, x x, y y = x y x y. () M (R) (A, B) = tr(a + B). (3) C([, ]) (f, g) = / f(t)g(t) dt. 7.5 (x, y), x, y ( ) ( ) () C + i i x =, y =. i i () C([, ]) (f, g) = f(t)g(t) dt F (x) = x, g(x) = e x. [ ] ( ) ( ) (3) R (x, y) = t 3 x y x =, y =. [ ] (4) M (R) (A, B) = tr(b t A) A =, B = [ ] (, ), (, ) V (, ) = (, ) + (, ) ) V 7.7 V x, y x + y + x y = x + y ( )

128 7 7.8 V x, y 4(x, y) = x + y x y + i x + iy i x iy 7.9 (x + y) (x y) x = y 7. V x, y V x y x y d(x, y) () (3) () d(x, y). d(x, y) = x = y. () d(x, y) = d(y, x). (3) d(x, y) d(x, z) + (z, y). 7. R 4 b a a =, a =, a 3 = c b a, b, c 7.. () R 4 a, a, a 3. a =, a =, a 3 =

129 7.. 3 i () C 3 a, a. a =, a = + i. i (3) R [x], x, x. (f, g) = f(t)g(t) dt 7.3 V 7.4 W [, π], (f, g) = π f(t) g(t) dt π S = {e inx n =, ±, ±, } 7.5 V {a,, a n } V () x V x = n i= (x, a i)a i. () x, y V (x, y) = n i= (x, a i)(a i, y). (3) x V x = n i= (x, a i). ( ) 7.6 V E = {a,, a n } f : V V E A = (a ij ) a ij = (f(a j ), a i )

130 V a,..., a m V (a, a ) (a, a m ) G = G(a,, a m ) = det.. (a m, a ) (a m, a m ) a,..., a m G(a,, a m ) 7.8 V K ϕ L(V, K) ϕ(x) = (x, a) (x V ) a V 7.9 V K S V S V S S = {x V y S, (x, y) = }. S V S S 7. V K X, W, W, W V () V = W W. () (W ) = W. (3) W W = W W. (4) (W + W ) = W W. (5) (W W ) = W + W.

131 S S () R 3 S = 3 ( ) i () C 3 S =, + i i ( ) (3) R [x] S = {x } ((f, g) = f(t)g(t) dt).) 7. A M m,n (K) rank A = r < n Ax =, 7. V K V (, ) V f( L(V )) f L(V ) (f(x), y) = (x, f (y)) (x, y V ) f f V y ϕ: V K, ϕ(x) = (fx), y) ϕ ϕ ϕ(x) = (x, a) a V 7.8) V y a f

132 6 7 A M n (K) t A A (A A a ij a ij ) A A K = R A = t A V K f L(V ) f : f f = f f. f : f = f. (K = R ) f : f = f. (K = R ) f : f f = f f = V. (K = R ) A M n (K), A : A A = AA A : A = A A : A = A. A : A A = AA = E. t AA = A t A = E () V K f, g L(V )

133 7.. 7 (.) f L(V ). (.) (f ) = f. (.3) (cf) = cf (c K). (.4) (f g) = g f. (.5) (f + g) = f + g. (.6) V = V. () A, B M n (K) (.) E = E (.) (A ) = A (.3) (ca) = ca (c K). (.4) (AB) = B A. (.5) (A + B) = A + B. (3) V K E V f L(V ). [f ] E = [f] E (4) V K E V f L(V ) f : [f] E : f : [f] E : f : [f] E :

134 8 7 f : [f] E : 7.6 A = 3i i + i 3 + i + 4i A i i + i + 3i i i + 3i 3 i i A = t A = t 3 i 4i = i 4i + i. i + i i i i 7.7 A M n (K) B = (A + A ), C = (A A ) B = (A + A ) = (A + A ) = (A + A) = B B C = (A A ) = (A A ) = (A A) = C

135 7.. 9 C B + C = (A + A ) + (A A ) = A A B C A = B + C = B + C, (B, B :, C, C : ) B B = C C (B B ) = (C C ). B B = C C. B B = C + C. C + C = C C. C = C. B = B. 7.8 A M n (K) A A, AA B = A + A. B = (A + A ) = A + A = A + A = A + A = B B C = AA C = (AA ) = A A = AA = C C

136 A : = t A = t A = A = A A = t AA = AA = A t A = A. A : = t A = t A = A = A A = t AA = AA = A t A = A. A : = t AA = A t A = E, t A = t A = A A = t AA = t AA = A t A = A t A = AA. A : = A = A = A A = AA = AA. A : = A = A = A A = ( A)A = A( A) = AA. A : = A A = AA (= E). 7. f L(V ) f(x) = f (x) (x V ). A M n (K) Ax = A x (x K n ). f f = f f f (x) = (f (x), f (x) = (x, f f (x)) = (x, f f (x)) = (x, f f(x) = (f(x), f(x)) = f(x).

137 7.. 3 f (x) = f(x). x V f (x) = f(x) f (x + y) = f(x + y) f (x) + f (y) + (f (x), f (y)) + (f (y), f (x)) = f(x) + f(y) + (f(x), f(y)) + (f(y), f(x)). (f (x), f (y)) + (f (y), f (x)) = (f(x), f(y)) + (f(y), f(x)) () f (x + iy) = f(x + iy) f (x) + f (y) i(f (x), f (y)) + i(f (y), f (x)) = f(x) + f(y) i(f(x), f(y)) + i(f(y), f(x)). (f (x), f (y))+(f (y), f (x)) = (f(x), f(y))+(f(y), f(x)) () () + () f (y), f (x)) = (f(y), f(x)) (y, f f (x)) = (y, f f(x)). y V f f (x) = f f(x) (x V ). f f = f f. f 7.3 i A = i i, A = i i i,

138 3 7 i i A 3 = i, A 4 = i i i i i i 7.4 f : C 3 C 3, x ix + 3y + ( i)z y iy + 3z z ( + i)x iz f 7.5 f L(V ) () f () (f(x), f(y)) = (x, y) (x, y V ). (3) f(x) = x (x V ). (4) E = {e,, e n } V {f(e ),, f(e n )} V (5) f 7.6 A = (a a n ) ( M n (K)) {a,, a n } K n 7.7 () ( ) ( cos θ sin θ cos θ sin θ cos θ sin θ ) sin θ cos θ ( θ < π)

139 () ( ) a b e iθ b e iθ a (θ :, a + b = ) 7.8 A ( M n (C)) A = B + ic (B, C M n (R)) () A : t B = B, t C = C. () A : t BB + t CC = E, t BC t CB = O. 7.9 A, B M n (R) A H = A + ib H = A E + (A B) A, B ( M n (R)), ( ) A B C = A + ib : : B A A ( M n (R)) Ã = ±t A A, B ( M n (R)) () AB () A = B A + B

140 A ( M n (C)) E + A = () A : = B = (E A)(E + A) : () A : = B = (E A)(E + A) : ( ) ( ) a b a b 7.34 b a b a 7.35 A ( M n (C)) () A = O = A = O. () C n x A x = = Ax = A,..., A m ( M n (C)) A + +A m = O A = = A m = O 7.37 f V x ( V ) p f p (x) = f(x) = 7.3 A ( M n (C)) Φ A (x) Φ A (x) = (x λ ) m (x λ k ) m k (λ i C)

141 A P λ... λ P AP =... λ k. O... λ k ( ) V U U U V = U U x V, x! U, x U : x = x! + x. p U : V V p U (x) = x! p U U 7. V U U u, u,..., u r p U : V V p U (x) = (x u )u + (x u )u + + (x u r )u r 7.3 p: V V (well defined)

142 ( ) T t T T = E = T t T 7.5 T = (u u... u n ) (u i n ) u, u,..., u n R n 7. n A () A R n U () A = T t T t T T = E r n r T () () : U u, u,..., u r u, u n,..., u n R n u r+, u r+,..., u n S = (u u... u n ), T = (u u... u r ) S ( ) E r T = S O ( ( )) t T = t E r S = O ( E r ) O t S.

143 { ui, i r Au i =, r < i n AS = T ( E r ) O A = AS t S = ( ( T E r )) (( O t S = T E r ) ) O t S = T t T. t T T = = ( E r ( E r = E r. ( ) ) O t E r SS O ) ( ) E r O O () () : T = (u u u r ) t T T = E r u, u,..., u r u, u,..., u n R n u r+, u r+,..., u n R n x x = x u + x u + + x n u n x! = x u + x u + + x r u r, x = x r+ u r+ + x r+ u r+ + + x n u n.

144 38 7 t T T = E r t T u k = e k Ax = T t T x = T x. x r = x! T = 6 A = T t T 7.39 R 4 3,, U 7. x V = y R 3 x + y + z = z R3 () V. () R 3 V p V A (3) R 3 V V p V A (4) A = T t T t T T = E, A = S t S t SS = E T, S

145 () V x x + y + z = x x = y x y a =, a = x x = y = x + y x y a, a V k a + k a = k = = k a + k a = k. k k k =, k = a, a a, a V -

146 4 7 u = a a =, b = a (a u )u = = u = b b = 3., u, u () 7. e =, e =, e 3 = p V (e ) = (e u )u + (e u )u = 3 u u = 7 6, 8 3 p V (e ) = (e u )u + (e u )u = u = 3 p V (e 3 ) = (e 3 u )u + (e 3 u )u = u = 4 3 u = 8 3 6,

147 A ( ) A = p V (e ) p V (e ) p V (e ) = x (3) V u 3 = y u u 3 =, u u 3 = z u 3 = x z = x y + z = x + y + z = x = y = z x = y = z = 3. V u 3 = 3 i =,, 3 (e i u 3 ) = 3 7. p V (e i ) = (e i u 3 )u 3 = u 3 =. 3 3 A A = ( ) p V (e ) p V (e ) p V (e ) = 3.

148 4 7 ) (4) T = (u u ( ) 3 T t T = ( ) = = A, ( ) ( ) 3 t T T = ( ) ( ) = 8 = = E 8 8 ] S = [u 3 ( ) ( ) S t S = 3 t SS = ( ) ) 3 ( S = = A 3 = ( ) ( ) 3 = = E 3 7. T, S A, A

149 x 7.4 R 3 P = y : 3x + y + z = z P 7.4 R 4 3,, U U

150

151 ( ) A Ax = λx, x x λ λ A x A λ 8. λ A A λe =. 8.3 ( ) n A n xe A A A xe A = 8.4 A A

152 A λ, λ,..., λ k x, x,..., x k 8. A = 4 4 det(a λe) = λ det(a λe) = det 4 λ 4 λ = ( λ)(4 λ)( λ) (4 λ) + 4( λ) λ = λ 3 + 5λ 8λ + 4 λ 3 +5λ 8λ+4 = (λ )(λ ) = λ =, λ = v v (A E)v = x y = 4 z x y z = y + z y = y + z y, z ) z

153 λ = x (x ) 8. ( ) 4 5 () () (3) (4) (5) (6) 3 (7) (8) (9) () A P A P AP A P AP xe A = xe P AP = E = P P xe =

154 48 8 xp P = xp EP = P (xe)p xe P AP = P (xe)p P AP = P (xe A)P. AB A B AB = A B xe P AP = P (xe A)P = P xe A P = P P xe A = P xe A = xe A. P xe A P xe A P = P xe A xe A = xe P AP = 8. A A = n A A = t A λ, λ V = {x R n Ax = λ x}, V = {y R n Ay = λ y} V V R n (Ax, y) = (x, t Ay)

155 λ V λ = {x R n Ax = λx} R n V λ x, y A(x + y) = Ax + Ay = λx + λy = λ(x + y) x y V λ k V λ x A(kx) = kax = kλx = λ(kx) kx V λ V λ R n V V R n V V V x V y (x, y) λ (x, y) = (λ x, y) = (Ax, y) = (x, t Ay) = (x, Ay) = (x, λ y) = λ (x, y) (λ λ )(x, y) =. λ λ (x, y) = 8.4 A A λe A

156 5 8 ( ) xe A = A a b A = b d ( ) ( a xe A = x b ( ) ( x a = x b x a b = b x d ) b d ) b d = (x a)(x d) ( b)( b) = x (a + d)x + ad b x (a + d)x + ad b = D D = (a + d) 4(ad b ) = a ad + d + 4b = (a d) + 4b D. D = a = d, b = D > A A 8.6 m m

157 () A n x R n (Ax, x) A () n A x R n (Ax, x) () λ A λ v Av = λv (Av, v) = (λv, v) = λ(v, v) = λ v. v > λ ( ) ( ) () A = x = A x xe A = = x(x ) = x A A

158 5 8 ( ) ( ) ( ) (Ax, x) = (, ) ( ) ( ) = (, ) = < x R n (Ax, x) 8.5 () 8. A = (a ij ) n x R n (Ax, x) a ij 8. A n A, B P AP = B P A B A B

159 n M n (R) A A, A B B A, A B, B C A C, 8.3 (3) (), () () A O = A = O. () A B A λe B λe. (3) A ce = A = ce. 8.4 A B A B ( ) ( ) ( ) ( ) A O B O A B,. O A O B O A O B 8.5 () B ()A B A B ()B AB BA n n

160 A = A λ,, λ = (A E)x = x = sp + tp. p =, p =. λ = (A + E)x = x = sp 3. p 3 =. p, p, p 3 A P = (p, p, p 3 ) P P AP = () () (3)

161 (4) 3 (5) (6) A A 8.7 ( ) A = 3 A λ xe A = x 3 x = x x 6 = (x 3)(x + ) λ = 3, ( ) λ = 3 (A 3E)x = x = s. ( ) λ = (A E)x = x = t. 3 ( ) ( ) P = P 3 AP =. 3

162 ( ) ( ) ( ) ( ) () () (3) (4) (5) (6) (7) a a a n a a n A =.... a nn 8.8 A, B M (R) A + B, AB ( ) a 8.9 A = n A n A n 8. A A A A

163 A A 8. A n λ A A λe 8.8 ( ) A = ( A) ( ) p = Ap = p. p ( ) R p = p, p R Ap = p + p ( ) A(p p ) = (p p ). P = (p p ) ( ) P AP =. 8.3 ( ) ( ) () 3 () 4 ( ) (3) 3 ( ) 5 4 (4) 9 7

164 A = 3 A (3 ) p = : Ap = p. q =, q 3 = Aq = p + q q 3, Aq 3 = p + q 3. A(p q q 3 ) = (p q q 3 ). ( ) B = B ( ) Q = ( ) Q BQ =

165 R = ( Q ) ( ) R P AP Q = R a R B ( ) ( ) ( a = Q B ( ) ( ) aq = Q BQ ( ) aq = Q BQ =. 8.4 () () (3) 3 Q ) 8. ( ) f(a) = O A f(x) 8. ( ) A = 3 A 3 + 3A

166 6 8 A x x + 5 A A 5E = O A x A x 3 + 3x = (x + )(x x 5) + 9x 5 A 3 + 3A = (A + E)(A A 5E) + 9A 5E = (A + E) O + 9A 5E ( ) ( ) = ( ) 3 9 =. 7 3 ( ) 8.5 A = 3 () A 4A E () A + 3A + 3E (3) A 3 + A + E 8.6 A = A 3 4A 3 8. k A k = O 8.7 A () A () A

167 n A λ B = A λe () B () B (3) B 8.9 A A a () P a AP =... () A = O a n a = a = = a n = 8.3 A A n O, A n = O B ( ) ( ) A C A O. O B O B. ( ) E R P = O E R = CB + ACB + A CB A n CB n

168 6 8 ( P A O ) ( C A P = B O ) O. B 8.4 A λ B λ B ( ) ( ) A C A O O B O B.. A k B k ( ) A λ E k O O B λ E k A λ E B λ E k 8.3 (A λ E) n O, (A λ E) n = O R = C(B λ E k ) + (A λ E k )C(B λ E k ) + + (A λ E k ) n C(B λ E k ) n, ( ) E k R P = O E k ( ) ( P A λ E k C A λ E k P = O B λ E k O ( ) ( ) P A C A O P =. O B O B ) O. B λ E k

169 A λ, λ,..., λ k A A A.... A i = λ i λ i.... A k λ i 8.3 n 3 λ, λ, λ 3 A A A A 3, A i = λ i λ i... λ i m J(α, m) α. α.... α α

170 ( ) P J P J AP = n A n J i 8.8 A P... J r 8. A ce λ A λ () λ v λ v v v := (A λe )v v λ () P v v P = (v v ). P P AP = ( λ λ )

171 () λ p, p v p p ( ) Av = λv v = ( ) v = A = λe A λa + λ E = O (A λe )v = (A λe ) v = (A λa + λ E )v = v λ. () Av = λv, Av = v + λv ) ( ) A (v ( ) λ v = v v λ 8. A = ( ) 3

172 66 8 ( ) ( ) ( ) ( ) v = v := (A E )v = ( ) P = ( ) P AP = 8.3 ) ( ) ( ) ( ) ( ) 3 () () (3) (4) 3 4 ( ) ( 6 ) ( ) 3 3 (5) (6) (7) ( ) ( ) 4 3 (8) (9) () () (3)

173 () () (3) 3 3 (4) (5) (6) (7) A A 8.35 n A, B tr(a) = tr(b) 8.36 A n n λ,..., λ n n n n I,, I n Ii = I i (i =,..., n), i j I i I j =, A = λ I + λ I + + λ n I n A A m 8.37 {a n } {b n } a =, b =, a n+ = 4a n b n, b n+ = a n + b n

174 68 8 ( ) ( ) ( ) a n a n a n () = A A A, b n b n b n a, b () a n, b n n (3) {a n }, {b n } a = b =, a n+ = a n b n, b n+ = 9a n + 5b n a n, b n n

175 69 ( ) 3. () A 3B + C =. 4 () () X = () X =, Y = 3. (3) X = , Y =.3 () AD = 3, AC = 6 8 DA = ( ) ( 7 9 () BC =, BD = (. ) 4, AE = ) ( ) 3 9 6, BE =, ( ) 3 4 3,

176 7 6 CB = (3) EC = 4 39, ED = 5, AE = ( ) BE = ( ) 3 4 3,.4 A, B n A = (a ij ), B = (b ij ) i j a ij =, b ij = AB (i, j) i j n k= a ikb kj = AB.5 (). () () A 4A+7E = O A 3 3A 7A+3E = (A + 5E )(A 4A + 7E ) A 3E = A 3E ( ) A 3 3A A + 3E =

177 () A =. () A = ( ) () CA BA =. 5 5 ( ) () (B 3C)A (C + B)A = ( ) 4 68 (3) (B + C)(C B)A = a = 5x + 6y + 5z b = 4x 5y 4z c = 4x + 8y + 6z. ( ) ( ) x y 3 () = x. () y 4 x 3 y x (3) y = 3 x. y 3 x 3 y y y 3 3 ( = 3 x x ).

178 7. () (x + y xy yz + 6xz) () (x + z xy yz). () ( ) ( ) A t a b a b A = = b a b a ( ) ( ) t a b a b AA = = b a b a () A = ae + bi, t A = ae bi. ( a + b a + b ( a + b a + b.3 A = (a ij ), B = (b ij ), C = (c ij ) () λ(ab) (i, j) ( n ) λ a ik b kj (λa)b (i, j) A(λB) (i, j) k= ) ).. (.3.) n λ(a ik )b kj (.3.) k= n a ik (λb kj ) (.3.3) k= (.3.)=(.3.)=(.3.3) () () (AB)C (i, j) ( n n ) a ik b kl c lj (.3.4) l= k=

179 73 A(BC) (i, j) n k= a ik ( n l= b kl c lj ) (.3.5) (.3.4)=(.3.5) () (3) (6).4 () (8) 5 4 () () ( ) ( ) ( ) a b a AB = =, c d c ( ) ( ) ( ) a b a b BA = = c d b =, c = () ( ) ( ) ( ) a b a AC = =, c d c ( ) ( ) ( ) a b c d CA = = c d c =, a = d ( ) a (3) (), () A = a

180 74 ( ) X XA = AX a A = a.6 ( ) ( ) 8 8 () AB =, BA = AB BA. 4 8 ( ) () (AB) = = A B..7 (4) A = (a ij ), B = (b ij ) tr(ab) tr(ba) AB, BA AB, BA AB (i, i) n k= a ikb ki, BA (i, i) n k= b ika ki ( n n ) tr(ab) = a ik b ki = = i= k= ( n n ) a ik b ki k= i= ( n n ) b ik a ki i= k= = tr(ba). (), (), (3).8 AB BA = re A, B r. tr(ab BA) = tr(re) tr(ab BA) = tr(ab) tr(ba) =.

181 75 (.9 () a b ) b a a b c () a b a. () A a a = a, A 3 3a 3a = 3a. () A n na n(n ) a = na.. X = E n X = (Y A)X = Y (AX) = Y E n = Y.. () t (A + t A) = t A + t ( t A) = t A + A = A + t A () t (A t A) = t A t ( t A) = t A A = (A t A) (3) (A + t A) + (A t A) = A. B = (A + t A)/, C = (A t A)/ (), () B C A = B + C (4) (3) B = (A + t A)/ =, C = (A t A)/ =

182 76.3 () X X = t X. X X = t X. X = t X = X X = O. () A = B +C, A = B +C B +C = B +C B B = C C X = B B = C C t X = t (B B ) = t B t B = B B = X X t X = t (C C ) = t C t C = C + C = X X () X = O B = B C = C.4 () ( x n ) = ( x)( + x + + x n ). () A n = O () E = (E A)(E + A + + A n ) (E + A + + A n )(E A) = E E A (3) () (E A) = E + A + + A n.5 () AB t (AB) = AB. A, B t (AB) = t B t A = BA. AB = BA. () AB = BA t (AB) = t (BA) = t A t B A, B t A t B = AB. t (AB) = AB AB.6 A m = O, B n = O () (A + B) m+n = m+n p= ( m + n p ) A p B m+n p

183 77 ( ) m+n p = (m+n)! (m+n p)!p!. p m B m+n p = O m < p m + n A p = O (A + B) m+n = O A + B () (AB) m = A m B m = O AB.7 A t A = A () t (A ) = t (A A) = t A t A = ( A)( A) = A A () t (A 3 ) = t (A A ) = t (A ) t A = (A )( A) = A 3 A 3 (3) n A n n A n.8 () A 3 =, A 3 = 3. 3 n n () n A n = n, n A n = n. n n.9 A =, A 3 =, A 4 =, A 5 =, A 6 = A 6k+ = A, A 6k+ = A,

184 78 A 6k+3 = A 3, A 6k+4 = A 4, A 6k+5 = A 5, A 6k+6 = E 3. k ( ) ( ).3 A α a b = A = c d ( A a = c ) ( b a d c ) ( ) b a + bc b(a + d) =. d c(a + d) bc + d c(a + d) = b(a + d) = α c = a + bc =, bc + d = a =, d = b(a + d) =.3 () A = (a ij ), B = (b ij ) A, B i > j a ij =, b ij = AB (i, j) n k= a ikb kj i > j k i a ik =, j + k n b kj = n k= a ikb kj = AB ().3 A, B.33 A n n B AB = BA = E n t (AB) = t (BA) = E n. t (AB) = t B t A, t (BA) = t A t B t B t A = t A t B = E n t A.34 () A 4A + 3E = O (A 4E )A = A(A 4E ) = 3E ( (/3)A+(4/3)E )A = A( (/3)A+(4/3)E ) = E

185 79 A () () A = 4 3 E 3 A.35 A C CA = AC = E n O = C(AB) = (CA)B = B B = O B () 7 8 () E =... O c O , ,

186 8.5 P ij (i) (j) AP ij = A(e e j e i e n ) (i) (j) = (Ae Ae j Ae i Ae n ) (i) (j) = (a a j a i a n ) i j AP i (c) A i c AP ij (c) A j i c ( ) () () (3) (4) 3 (5) (6) a 5 a + a + 6 a = 6.8 a = a = 3

187 8.9 () () x y z = 7 + λ 6 + λ (λ, λ w { x = z w (z, w y = 6z + 7 (3) x y = z { (4) + λ x = z y = z + (λ ) (z. a b a b b a b a A = c d c d d c d c A 4 (a, b) (, ) (c, d) (, )

188 b a 3 4 a b b = b = a = a (, 3, 3 ) (, (3 4 ) (3, 3 )

189 83 ( ) ( 3 ) 5 5 (3 5 ) (3 ) x y z = + α w ] α

190 84.3 B = a a (3 ) a ( ) a ( ) a ( 3 ) a 3 a = a a a 3 a

191 () 7 () 6 (3) (4) (5) (6) 4 (7) 8 3. () () (3) (4) 3.3 () () + (3) () a a a 33 a 44 () a a a 33 a 44 (3) a a a 33 a 44 +a a 3 a 34 a () k n a a a nn () n ( ) n/ a,n a,n a n,, n ( ) (n )/ a,n a,n a n,. ( ) (n )n/ a,n a,n a n, 3.8 () 4 () (3) (4) 3

192 () kl () klm (3) 3. () () (3) (4) 7/36 (5) (6) 56 (7) 6 (8) 4 (9) () 6 () 7 () 3 (3) 3. () A () A (3) A (4) A 3. () k A () k 3 A 3.4 () 5 () 7 (3) 43 (4) (5) 46 (6) 3 (7) 3 (8) (9) 3 () 63 ( ) ( ) 3.6 () 3 () 3 (3) 7 8 ( ) (4) (5) (6) (7) () x = 3, x = 3, x 3 =. () x = 4, x = 3, x 3 =. (3) x =, x = 5, x 3 = 5. (4) x = 3 5, x = 7 5, x 3 = p = (a, b), q = (x, y) θ p q = p q cos θ cos θ = p q p q.

193 87 4. p q = (a, b) (x, y) = ax + by, p = a + b, q = x + y cos θ = ax + by a + b x + y 4. p = (a, b, c), q = (x, y, z) θ p q = p q cos θ 4. cos θ = p q p q. p q = (a, b, c) (x, y, z) = ax + by + cz, p = a + b + c, q = x + y + z cos θ = ax + by a + b + c x + y + z

194 p = (a, b), q = (x, y) p q = (a, b) (x, y) = (a x, b y) p q = (p q) (p q) = (a x, b y) (a x, b y) = (a x) + (b y). 4.4 p = (a, b, c), q = (x, y, z) p q = (a, b, c) (x, y, z) = (a x, b y, c z) p q = (p q) (p q) = (a x, b y, c z) (a x, b y, c z) = (a x) + (b y) + (c z). 4.5 p = (a, b), q = (x, y) 4.7 p q = (a, b) (x, y) = ax + by = 4.6 p = (a, b, c), q = (x, y, z) 4.7 p q = (a, b, c) (x, y, z) = ax + by + cz = 4.7 p = (, ), q = ( 3, 3) p q θ 4.7 cos θ = p q (, ) ( 3, 3) 3 3 = p q (, ) ( 3, 3) = 6 = =. 6 π 4 (= 45 )

195 p = (,, ), q = (, +, ) p q θ 4.7 cos θ = p q p q = = = (,, ) (, +, ) (,, ) (, +, ) ( ) + ( + ) + 3 ( ) + ( + ) = 3. π 6 (= 3 ) = S P k O P k P i, P k P j x i x k y i y k S = x j x k y j y k = x i x k y i y k x j x k y j y k x k y k = x i y i. x j y j i, j, k,, A(, 3), B(, ), C(3, ) 3 = = 4. 3

196 9 ABC 4. 3 P (x, y ), P (x, y ), P 3 (x 3, y 3 ) 4.9 x y x y = x 3 y 3 3 P (x, y ), P (x, y ), P 3 (x 3, y 3 ) 4.9 x y x y = x 3 y 3 4. () 4.5 (3, 4) (5, ) θ cos θ = (3, 4) (5, ) (3, 4) (5, ) 5 48 = 5 3 = θ cos θ = 33 65

197 9 () ( 4.3) 3x 4y+5 = 3x + 4y 5 = m = , n = , d = x 4y + 5 = 3 5 x y =. 5x + y 6 = m = , n = , d = x + 3 y = (3) O (md, nd) 3x 4y + 5 = ( 3, 4) 5 5 5x + y 6 = ( 5, ) = (, 4) x 4y + 5 =, 3x 4y 5 = 3 5 x y =, 3 5 x 4 5 y = 3 4.3, 3 3 =

198 9 4.4 P (x, y ), P (x, y ) P(x, y) 3 P, P,P 4. P x y x y = x y 4.5 P P = (x x, y y, z z ) 4. x x x x = y y y y = z z z z 4.6 cos θ = e l e l cos η = f l f l = (, ) (l, m) = l, = (, ) (l, m) = m (cos) l x, y, z α, β, γ l + m + n = l = cos α, m = cos β, n = cos γ l x, y, z α, β, γ l = cos α, m = cos β, n = cos γ

199 a = (, 3, ), b = (,, 3) 4.5 ( ) 3 a b = 3, 3, 3 = ( 9, 4 3, 6) = ( 9, 7, 6). 4. () 4.7 a, b e e e = e e e e = e e = f, g () e f x e, y f z e f g e f = f e 4.7 (3) (4) 4. 3 P (x, y, z ), P (x, y, z ), P 3 (x + x, y + y, z + z ) S OP OP 4.5 ( ) OP y z OP = y z, z x z x, x y x y

200 94 S = y z z x x y + + y z z x x y 4. O 3 P (x, y, z ), P (x, y, z ), P 3 (x 3, y 3, z 3 ) OP i (i =,, 3) 6 V OP OP S S = OP OP OP OP OP 3 θ h h = OP 3 cos θ V V = Sh = OP OP OP 3 cos θ = OP OP OP 3 ( ) y z = y z, z x z x, x y (x 3, y 3, z 3 ) x y = y z y z x z x 3 + z x y x y 3 + x y z 3 = y z y z x 3 x z x z y x y 3 + x y z 3 x y z = x y z x 3 y 3 z 3

201 O 3 P (x, y, z ), P (x, y, z ), P 3 (x 3, y 3, z 3 ) O 3 P, P, P 3 OP i (i =,, 3) 6 V 4. x y z V = x y z = x 3 y 3 z 3 V, O 3 P, P, P 3 OP i (i =,, 3) 6 V OP OP h = OP 3 cos θ cos θ P 3 3 P, P, P a = (, 3, ), b = (,, 3), c = (,, ) 4. 6 V 3 V = 3 = =. 5. k, k, k 3 (), k 3. (), k,, k,

202 96, k 3. (3), k,, k, 3, k (4), k, 7, k 9. (5) a, k. (6), k + k,, k 3. (7), k + k, 5, k 3.

203 97 (8), k,, k, 5, k () P = P AP = 3. 4 () P = P AP = () P = 4 P AP =. 3 () (3) a = b a b a P = P AP = a. b a b 5.4 ( / / ) ( ) () P = / / P AP =. 4

204 98 ( / 5 / ) ( ) 5 () P = / 5 / P 3 AP = / / () P = / / P AP =. 4 /3 /3 /3 () P = /3 /3 /3 P AP =. /3 /3 / ( ) 8 () P = P AP = 3 ( ). 3 ( ) ( ) A n n = P P = 8 n + ( 3) n+ 8 n + 8 ( 3) n. ( 3) n 5 3 n + ( 3) n+ 3 n + 8 ( 3) n ( ) ( ) () P = P AP =. 5 A n = P ( ) ( ( ) n P = ( ) n + 5 n 5 n ( ) n 5 n ) ( ) n 5 n ( ) n + 5 n 5.7 a + b, (a + b), a b, (a b). 5.8 A x Ax = A x = A Ax =

205 λ A x Ax = λx A 3 x = A (Ax) = A (λx) = λa(ax) = λ Ax = λ 3 x. A 3 = A A 3 x = Ax = λx. x λ 3 = λ λ =,, 5. P P AP = a... a n i a i.) B = P a... an P B = a P... P = P = A a... a n an P 5. () xe A x n (x a )(x a ) (x a nn ) x n

206 n i= a ii tra = n i= a ii x n tra () Φ A (x) Φ A () Φ A () = A Φ A (x) ( ) n A 5. () () 5.3 Φ P AP (x) = xe P AP = P (xe)p P AP = P xe A P = xe A P P = xe A = Φ A (x) Φt A(x) = xe t A = t (xe) t A = t (xe A) = xe A = Φ A (x) Φ A (x) = xe A = xe A xe A = Φ A (x) Φ A (x)

207 5.4 { x n = ( 3 n 5( 4) n )/7 () y n = ( 3 n 5( 4) n )/7 x n = ( + 5 n )/ () y n = ( + 5 n )/ z n = (5 n 3)/ 5.5 x n = (( ) n + 3 n )/ { x = (8e t 3e 3t )/5 () x = (3e t 3e 3t )/5 x = (e t e 5t )/3 () x = (e t 7e t + 8e 5t )/3 x 3 = (e t 7e t + e 5t )/3 5.7 y = (6e t 9e t + 5e t )/. 6. A, B n m k K A + B ka n m Mat(n, m) (V) (V8) A = (a ij ) A ( a ij ) 6. K n () (a n ) ( a n ) 6.3 f, g C(a, b) f + g [a, b] f + g C(a, b) k kf C(a, b) x [a, b] f C(a, b) f ( f)(x) = f(x)

208 6.4 K n [x] K n x (a + a x + + a n x n ) + (b + b x + + b n x n ) = (a + b ) + (a + b )x + + (a n + b n )x n k(a + a x + + a n x n ) = (ka ) + (ka )x + + (ka n )x n K n [x] a +a x+ +a n x n ( a ) + ( a )x + + ( a n )x n 6.5 V = { x K m Ax = } x, x V, k K A(x + x ) = Ax + Ax =, A(kx ) = kax = V K m 6.6 (i, j) n m E i,j { E i,j i n, j m } Mat(n, m) i n, j m k i,je i,j = O n,m O n,m n m k i,j = i, j { E i,j i n, j m } A = (a i,j ) Mat(n, m) A = i n, j m a i,je i,j { E i,j i n, j m } { E i,j i n, j m } Mat(n, m) Mat(n, m) nm