II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F 6, b F 6, ab = F 6. 2. () x R[x] = {f(x) = a n x n + a n x n + + a 0 n N, a 0,, a n R }., R[x] x 2 R[x]/(x 2 ) = {ax + b a, b R },., R[x]/(x 2 )., 0 0 ax + b R[x]/(x 2 ), cx+d R[x]/(x 2 ), (ax+b)(cx+d) = R[x]/(x 2 ). (2), p, q R, R[x], x 2 + px + q R[x]/(x 2 + px + q)., R[x]/(x 2 + px + q), p, q.
II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I. 6. 4 4 E, 4 4 I, J, K. 0 0 0 0 0 0 0 0 0 0 0 0 I = 0 0 0, J = 0 0 0 0 0 0, K = 0 0 0 0 0 0. 0 0 0 0 0 0 0 0 0,. ()., A, A (, A. ) t A. I 2 = J 2 = K 2 = E, IJ = JI = K, JK = KJ = I, KI = IK = J, t I = I, t J = J, t K = K. (2) A = we + xi + yj + zk, (x, y, z, w R), ν(a) = w 2 + x 2 + y 2 + z 2 R., (),. A t A = t AA = ν(a)e (3) A = we + xi + yj + zk O, A. O.,, O, I. m m A = (a ij ), tr A = m i= a ii, tr A A (trace). 7. m n A n m B, tr(ab) = tr(ba).
II 3 8. m n, m n A n m B, AB = I m, BA = I n., m N, m m I m. 9. m m A, B, AB BA = I m. N n = O n N N (nilpotent matrix). 0. m m A = (a ij ), i j, a ij = 0., A.. m m N, A = I N (,. )., A = I N A = (I N) N. 2. N, e N = n=0 N n n! = I + N + N 2 2! + N 3 3! + +. ( N,. ),. () e O = I. (2) N, N,, (N + N ), e N+N = e N e N., N, N, NN = N N. (3) N, e N, (e N ) = e N. (4) N, A(t) = e tn, (e tn ) = Ne tn = e tn N., A(t), A (t), A (t) = lim h 0 A(t + h) A(t) h. (, A(t) A(t) = ( a ij (t) ), A(t + h) A(t) = {( ) ( )} a ij (t + h) a ij (t) h ( h ) aij (t + h) a ij (t) = h, A (t) = ( a ij (t)). )
II 4 3. rank. () (4) 0 2 2 0, (2) 0 3 4 2 2, (5) 2 0 2 2 3 3 0 0 2 4, (3) 2 2 2 0 3 2 0 3. 2 0 2 4 0 0 0 2 2, 4.. 3 4 () 0 3, (2) 2 5 4 (4) 0 0 0, (5) 0 0, (3) 0 0 2 3 2 0. 2 6 6 2 7 6, 2 7 7 5.. 2 0 0 2 0 () A 4 = 0 2, (2) B 4 = 0 0 2 2 0 0 2 (3) D 4 = 0 2 0, (4) F 4 = 0 0 2 2 0 0 0 2 0 0 (5) A 5 = 0 2 0, (6) B 5 = 0 0 2 0 0 0 2 2 0 0 0 2 0 0 (7) D 5 = 0 2, (8) E 6 = 0 0 2 0 0 0 0 2 2 0 0 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 0 0 2,, 2 0 0 0 2 0 0 0 2 0 0 0 2 2 0 0 0 2, 2 0 0 0 0 2 0 0 0 0 2 0 0 0 2 0 0 0 0 2 0 0 0 0 0 2.
II 5 6.. 2 3 4 5 6 7 8 () 9 20 2 22, (2) 23 24 25 26 x 0... 0. 0 x... (3).......... 0, (4) 0... 0 x a 0 a... a n 2 x + a n (5) a 0! a!. a n! (a 0 )!... (a 0 n)! (a )!... (a n)!.. (a n )!... (a n n)! a b c d a 2 b 2 c 2 d 2 a 3 b 3 c 3 d 3,... x x 2... x n x 2 x 2 2... x 2 n... x n 2 x2 n 2... xn n 2 x n x n 2... x n n, (, n a 0 < a < < a n. ). 7. m m A n n B,. ( ) O A det = ( ) mn det A det B B O 8. n n A, B,. ( ) A B det = det(a B) det(a + B) B A 9. A, B, C, D n n.,, A, ( ) A B det = det A det(d CA B) C D, D, ( A det C ) B = det D det(a BD C) D. 20. A., A,, A, det A = ±.
II 6 2. a... 0 a 22.... A = 0 0........ 0 0... 0 a nn,., A, A A. 22. n n A, A Ã.,. () A, Ã = (Ã ). (2) n n A, B, ÃB = BÃ. (3) Ã = (det A) n 2 A. 23.. x y + z = 2 x y + z = 2 () 2x 2y + z = 3 (2) 2x 2y + z = 2 x + y + 2z = x + y + 2z = 3 x + y + 4z u + 2v = 3y + 3z 4u + 4v = 0 x + y 2z + u + 3v = (3) (4) 2x y + 2z + 2u + 6v = 2 2x y + 5z + 6u + 2v = 8 3x + 2y 4z 3u 9v = 3 2y + 2z + 2u + 5v = 7 24. V, W R n R n.,. () V W R n, V W = {u R n u V,, u W }, V W. (2) V v V W w W, v + w R n, V + W = {v + w R n v V, w W }, V + W. (3) V,, W R n, V W = {u R n u V,, u W }, V W
II 7 25. n n M n (R), V, V 2, V 3, V 4., V, V 2, V 3, V 4,,,,. () V = {X M n (R) tr X = 0 } (2) V 2 = {X M n (R) det X = 0 } (3) V 3 = {X M n (R) t X = X } (4) V 4 = {X M n (R) t XX = I } 26. R 3.,,,,., (3), a, b, c R,. 5 6 0 0 (), 3, 3 (2) 2, 5, 0 (3) 0 2 3 4 7 a a 2, b b 2, 27..,. () u, u 2,, u n R n,., k n, n k i < i 2 < < i k., k u i, u i2,, u ik. (2) u, u 2,, u n R n,., k n, n k i < i 2 < < i k., k u i, u i2,, u ik. (3) u, u 2,, u n R n,., R k ( ) v, v 2,, v n R k, u i v i w i =, w, w 2,, w n R n+k. u i v i c c 2 R n+k (4) u, u 2,, u n R n,., R k ( ) v, v 2,, v n R k, u i v i w i =, w, w 2,, w n R n+k. u i v i R n+k 28. u, u 2,, u n R n,.,. () u 2u 2 + u 3, 2u u 3, u + u 2 + u 3. (2) u + u 2, u 2 + u 3,, u n + u
II 8 29. α, α 2,, α n R,., x n e α x, e α 2x,, e α nx, R. 30. n N, n, V n = { f(x) = a 0 + a x + + a n x n a 0, a,, a n R }.,. () {, x, x 2,, x n }, V n. (2) c R, {, (x c), (x c) 2,, (x c) n } V n., () (2). 3. 30, V 3 3., V 3,. { } () W = f(x) V 3 f(x)dx = 0 (2) W 2 = { f(x) V 3 f() = f( ) = 0 } 32. V, W ( R ).,. () f : V W, g : V W, f + g : V W. (2) a R, f : V W, af : V W. (3), U ( R ), f : V W, g : U V,,, f g : U W. 33. 2,, f(x) V 2, V 2 = { f(x) = a 0 + a x + a 2 x 2 a 0, a, a 2 R } D(f)(x) = (x ) df dx (x) D : V 2 V 2.,. () D. (2) V 2 {, x, x 2 } D ˆD. (3) V 2 {, (x ), (x ) 2 } D Ď.
II 9 ( R ) V, V R, V = { f : V R f }., V, V., f, g V, a R, (f + g)(u) = f(u) + g(u), u V (af)(x) = a f(u),, V ( R ). V V. 34. V n ( R ), V {e, e 2,, e n },, V u V, u = a e + a 2 e 2 + + a n e n, a, a 2,, a n R., f i (u) = f i (a e + a 2 e 2 + + a n e n ) = a i V f i : V R, (i =, 2,, n)., f i, u V, u e i. f i, (i =, 2,.n),. () i =, 2,, n, f i V. (2) {f, f 2,, f n } V. 34 V {f, f 2,, f n }, V {e, e 2,, e n }. 35. V, W ( R ), ϕ : V W., f : W R, f ϕ : V R, f W, ϕ (f) := f ϕ V ϕ : W V. () ϕ : W V. (2) dim R V = 2, dim R W = 3, V {e, e 2 } W {f, f 2, f 3 }., V {e, e 2 } V {e, e 2 }, W {f, f 2, f 3 } W {f, f 2, f 3 }., ϕ : V W {e, e 2 }, {f, f 2, f 3 } A, ϕ : W V {f, f 2, f 3 }, {e, e 2 } B.
II 0 V, W f : V W, Ker f = {u V f(u) = 0 } Im f = {f(u) W u V } Ker f V, Im f W,, f (Kernel) (Image).,, V, W. 36. n n M n (R), A M n (R), ad A (X) = AX XA, (X M n (R)) ad A : M n (R) M n (R).,. () ad A : M n (R) M n (R). (2) i, j =, 2,, n, i j, 0 E ij, Ker ad Ei j, Im ad Eij. 37. 2 2 ( a A = c ) b d, ad A (X) = AX XA 2 2 M 2 (C) ad A : M 2 (C) M 2 (C). ( ad A, 36. ),. () M 2 (C) {E, E 2, E 2, E 22 } ad A âd A., ( ) ( ) ( ) ( ) 0 0 0 0 0 0 E =, E 2 =, E 2 =, E 22 = 0 0 0 0 0 0. (2) Z A = {X M 2 (C) AX = XA }, Z A M 2 (C),. 38. A l m, B m n,. rank A + rank B m rank(ab) min{rank A, rank B} 39. A rank A = r m n., rank B = r m r B rank C = r r n C, A = BC.,, A,, rank A = r.
II 40. A n n, A, (), (2) n n B, C, A = BC. () B. (2) C 2 = C 4. n n A M n (R), A 2 = A,. rank A = tr A 42. n n A = (a ij ) a ij 0, i =, 2,, n, n j= a ij =. ( A. ),. () A. (2) λ A, λ. 43. n. () A = 5 2 4 2 0 2, (2) B = 2 4 0 0 3 3 0 3 0 0 0 6 0 0 5 44. n n A M n (R),. () A., t AA = I. (2) A a, a 2,, a n R n,, i = j a i, a j = 0, i j. (3) u, v R n, Au, Av = u, v. (4) u R n, Au = u. (5) {e, e 2,, e n } R n, {Ae, Ae 2,, Ae n } R n.
II 2 45. 2 2 A, R θ, T θ., ( ) ( ) cos θ sin θ cos θ sin θ R θ =, T θ = sin θ cos θ sin θ cos θ. 46. U det U,., z C, det U = z U. 47., Gram-Schmidt, R 3. 0 0 0 (),, (2),, (3),, 0 0 0 48. 2 V 2, f(x), g(x) = f(x)g(x)dx, f(x), g(x) V 2., {, x, x 2 } V 2, Gram- Schmidt, V 2., U, U V U, V = {u U v V, u, v = 0 } U V, V. 49. V R n, R n.,. () R n, R n = V V., u R n, u = u + u 2, u V, u 2 V. (2) u R n, () u V u V, V, u., v V, u v u u. 50. V, V, V 2 R n, R n.,. () (V ) = V (2) (V + V 2 ) = V V 2 (3) (V V 2 ) = V + V 2
II 3 5. n n M n (R), X, Y = tr( t XY ), X, Y M n (R),.,. () X, Y, X, Y X = (x ij ), Y = (y ij ). (2) V M n (R),., V. (3) V V. 52. V, R., u V, f u (v) = u, v, v V f u : V R. () u V, f u V., f u : V R. (2) V, u V, f u V f : V V,., f,,,. 53. A m n.,. () u R n, v R m, Au, v R m = u, t Av R n., R m, R n,,, R m,, R n. (2) Ker A, Im A,, (a) (Ker A) = Im t A (b) (Im A) = Ker t A. n n A, u R n, u 0 = t uau > 0,., n n B, v C n, v 0 = t vav > 0,.
II 4 54. n n A, A, A. 55. A, B n n, 0 < λ, µ R, λa + µb,. 56. A, B n n, tr(ab) > 0. 57. A n n, B n n., n n P, t P AP,, t P BP.