, 5, ;,,,,,, ; ; 4,,, ; () 1345, 2,,,,,,,, 2014 2
1 1 11 1 111 1 112 2 113 Cramer 3 12 3 121 3 122 4 123 4 13 5 131 5 132 13 133 13 134 Cramer 14 135 16 14 17 15 20 16 () 27 2 30 21 31 211 31 212 31 213 32 214 34 215 34 216 35 217 35 218 36 219 36
iv 22 37 221 37 222 37 223, 37 224, 37 225 38 226 38 23 38 231 38 232 39 233 41 234 43 235 44 236 44 24 45 25 48 26 () 57 3 60 31 61 311 61 312 62 313 63 314 63 32 64 321 64 322 64 323 65 33 65 331 65 332 67 333 () 69 334 70 335 73 34 75 35 88
v 36 () 99 4 101 41 102 411 102 412 102 413 103 414 103 415 103 416 Gram-Schmidt 104 42 104 421 104 422 104 423 105 424 105 425 105 426 105 43 105 431 : 105 432 : n 109 433 : 111 434 : 112 435 : 114 436 : 115 44 116 45 122 46 () 135 5 138 51 139 511 139 512 139 513 139 514 141 515 142 52 142 521 142
vi 522, 142 53 143 531 143 532 143 533 144 534 146 54 149 55 156 56 () 168 1 171 2 190
1 1 2 () 3 (Cramer) 111 1 () 11
2 1 a 11 a 12 a 1n a 22 a 2n a nn = a 11 a 21 a 22 a n1 a n2 a nn = a 11 a 22 a nn 2 λ 1 λ 2 a 2n λ n a n2 a nn = a 11 a 1n 1 λ 1 a 21 λ 2 λ n = ( 1) n(n 1) 2 λ 1 λ 2 λ n 3 n(n > 1) (Vandermonde) 1 1 1 x 1 x 2 x n V n = x 2 1 x 2 2 x 2 n = x n 1 1 x n 1 2 x n 1 n 4 A O B 1j<in (x i x j ) O B = A = A B, O B A = A O ( 1)km B = ( 1)km A B A k, B m 112 1 (), (), a i1 A i1 + a i2 A i2 + + a in A in = D (i = 1, 2,, n) a 1j A 1j + a 2j A 2j + + a nj A nj = D (j = 1, 2,, n) 0, 2 () () a i1 A j1 + a i2 A j2 + + a in A jn = 0 (i j)
12 3 a 1i A 1j + a 2i A 2j + + a ni A nj = 0 (i j) 113 Cramer 1 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1, a 21 x 1 + a 22 x 2 + + a 2n x n = b 2, a n1 x 1 + a n2 x 2 + + a nn x n = b n D 0, x j = D j D (j = 1, 2,, n), D j D j a 1j, a 2j,, a nj :, 0 a 11 x 1 + a 12 x 2 + + a 1n x n = 0, a 21 x 1 + a 22 x 2 + + a 2n x n = 0, 2, a n1 x 1 + a n2 x 2 + + a nn x n = 0 0 D 0 121 12,, (),, 1 n, n, j 1 j 2 j n, (), j 1 j 2 j n, τ(j 1 j 2 j n ), 2 n a 11 a 12 a 1n a 21 a 22 a 2n D = det(a ij ) = = ( 1) τ(j1j2 jn) a 1j1 a 2j2 a njn, j 1j 2 j n a n1 a n2 a nn
4 1 1, 2,, n (n! ), τ(j 1 j 2 j n ) j 1j 2 j n j 1 j 2 j n 122,, a b 2, = ad bc, 2, c d,, det(a ij ) 2 2 D 2, ad bc, 2 123 n det(a ij ) a ij n 1 a ij, M ij A ij = ( 1) i+j M ij a ij a ij, 1 2 3 a b c 0 1 6 0 1 6 4 3 7 4 3 7, n i j, a i1 A j1 + a i2 A j2 + + a in A jn = 0 j,, n a 11 a 12 a 1n a i1 a i2 a in a i1 a i2 a in ( j i ) j, a 11 a 12 a 1n a a i1 A j1 + a i2 A j2 + + a in A jn = i1 a i2 a in = 0 a i1 a i2 a in
13 5 131 13 (), ; () ; (); n 1 n 1 n D n = 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 n 1 0 0 0 0 0 0 0 0 n n,, :, D n = ( 1) τ(n 1,n 2,,2,1,n) a 1,n 1 a 1,n 2 a n 1,1 a nn = ( 1) (n 1)(n 2) 2 n! 2 () 2 n D n = x y 0 0 0 0 x y 0 0 0 0 0 x y y 0 0 0 x 1 x y 0 0 0 y 0 0 0 0 0 x y 0 0 x y 0 0 0 D n =x + ( 1) n+1 y 0 0 0 x y 0 0 0 y 0 0 0 0 0 x 0 0 0 x y =x n + ( 1) n+1 y n
6 1 (), () 3 3 : 0 1 2 4 1 2 3 4 (1) 2 0 1 1 1 3 5 2 ; (2) 2 3 4 1 3 4 1 2 2 1 0 5 4 1 2 3 (1) 1: 1 3 5 2 1 3 5 2 = 0 1 2 4 2 0 1 1 = 0 1 2 4 0 6 11 5 2 1 0 5 0 7 10 9 1 3 5 2 1 3 5 2 = 0 1 2 4 0 0 1 19 = 0 1 2 4 0 0 1 19 = 57 0 0 4 19 0 0 0 57, () a 11 0, 1, 1 0 1 0, 2: 0 1 2 4 1 2 4 1 2 4 0 6 11 5 = 6 11 5 = 0 1 19 = 57 1 3 5 2 0 1 1 4 1 1 4 0 3 0 ===== r4 r2 r 2+2r 3, a 11 0, (), 1 2 3 4 1 2 7 (2) 1 = 0 1 2 7 0 2 8 10 = 2 8 10 7 10 13 0 7 10 13 1 2 7 = 0 4 4 = 160 0 4 36
13 7 2, 2, 3, 4 1 1 2 3 4 1 2 3 4 =10 1 3 4 1 1 4 1 2 = 10 0 1 1 3 0 2 2 2 1 1 2 3 0 1 1 1 1 2 3 4 =10 0 1 1 3 0 0 4 4 = 160 0 0 0 4, 4 : ax + by ay + bz az + bx x y z (1) ay + bz az + bx ax + by = (a 3 + b 3 ) y z x ; az + bx ax + by ay + bz z x y bc a a 2 1 a 2 a 3 (2) ac b b 2 = 1 b 2 b 3 (a, b, c 0) ab c c 2 1 c 2 c 3 (1), x ay + bz az + bx y ay + bz az + bx = a y az + bx ax + by + b z az + bx ax + by z ax + by ay + bz x ax + by ay + bz x y z y z x x y z = a 3 y z x + b 3 z x y = (a 3 + b 3 ) y z x ; z x y x y z z x y bc a a 2 (2) ac b b 2 = 1 abc a 2 a 3 1 a 2 a 3 ab c c 2 abc abc b 2 b 3 = 1 b 2 b 3 abc c 2 c 3 1 c 2 c 3,, n (), 2 n
8 1 (2) a, b, c 0, 1 a 2 a 3 1 b 2 b 3 1 c 2 c 3 5 n (1) D n = 1 2 3 n 2 3 4 1 n 1 n 1 n 2 n 1 2 n 1 ; (2) D n = a 1 + b 1 a 1 + b 2 a 1 + b n a 2 + b 1 a 2 + b 2 a 2 + b n a n + b 1 a n + b 2 a n + b n (1) i ( 1) i + 1 (i = n 1, n 2,, 1), 2, 3,, n 1, D n = 1 2 3 n 1 1 1 1 n 1 1 1 n 1 1 1 n 1 1 = n(n + 1) 2 2 3 n 0 1 1 1 n 0 1 1 n 1 0 1 n 1 1 = n(n + 1) 2 1 1 1 n 1 1 n 1 1 n 1 1 n 1 n 1 1 ( 1), 1, 2,, n 2 n 1, ( 0) D n = n(n + 1) 2 1 1 1 n 0 n n n 0 n = n(n + 1) 2 1 1 1 n n = n(n + 1) 2 ( 1) (n 1)(n 2) 2 ( n) n 2 ( 1) = ( 1) (n 1)n 2 n(n + 1) 2 n n 2
13 9 (), () (),, (2) n = 1, D 1 = a 1 + b 1 ; n = 2, D 2 = (a 1 + b 1 )(a 2 + b 2 ) (a 1 + b 2 )(a 2 + b 1 ) = (a 1 a 2 )(b 1 b 2 ); n 3, ( 1),, a 1 + b 1 a 1 + b 2 a 1 + b n a 2 a 1 a 2 a 1 a 2 a 1 D n = a 3 a 1 a 3 a 1 a 3 a 1 = 0 a n a 1 a n a 1 a n a 1, a 1 + b 1, n = 1, D n = (a 1 a 2 )(b 1 b 2 ), n = 2, 0, n 3, 4 n, n 6 n 5 3 0 0 0 0 2 5 3 0 0 0 D n = 0 2 5 3 0 0 0 0 0 0 2 5 D n = 5D n 1 2 3D n 2 = (2 + 3)D n 1 2 3D n 2, D n 2D n 1 = 3(D n 1 2D n 2 ) = 3 2 (D n 2 2D n 3 ) ( ) = = 3 n 2 (D 2 2D 1 ) = 3 n 2 5 3 2 5 10 = 3 n
10 1 7 n cos θ 1 0 0 0 1 2 cos θ 1 0 0 D n = 0 1 2 cos θ 0 0 0 0 0 1 2 cos θ = cos nθ n = 1, ; cos θ 1 n = 2, D 2 = 1 2 cos θ = 2 cos2 θ 1 = cos 2θ, ; n 1, D n 1 = cos(n 1)θ D n n cos θ 1 0 0 1 2 cos θ 0 0 D n = ( 1) n+(n 1) 0 1 0 0 + ( 1) 2n 2 cos θ D n 1 0 0 1 1 = ( 1) n+(n 1) D n 2 + ( 1) 2n 2 cos θd n 1 = cos(n 2)θ + 2 cos θ cos(n 1)θ = cos(n 2)θ + (cos nθ + cos(n 2)θ) = cos nθ 8 1 1 cos θ, D n n n x 1 x 1 D n = x 1 a n a n 1 a 2 a 1 + x 1 i x i 1 (i = n, n 1,, 2), 1 1