2.1.1 ( ) U V. U λ F, α, β U, α + β U λα U. 0 (, ),. ( 2 ): (1) : U V, W U, W V ; (2) ( ),, ;, U W U W., U W (?). V U W, U W?, α + β, α U, β W.

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1 ,. Ax = b, ( ),,.,, Ax = b ( ) F n F m σ : x Ax b F m., Ax = 0 ( ) σ : x Ax. σ : x Ax (T1) ( ) σ(x + y) = σ(x) + σ(y); (T2) ( ) σ(ax) = aσ(x), a F., τ : F n F m, A m n σ(x) = Ax, x F n., b F m σ Ax = b., A (T1) (T2) ( : ),. : C R 2, 1, R C., ( ) V, U V. U V, U V., {0},.. 0 ( 0 ). V, R C, R C, R C, R C.,, 1.4.1, V = { } R, V, R, V R F[x] n F[x] n. [a, b], R[x] R[x] n [a, b] C[a, b],, C[a, b]. V U, U V ;, :,, 1.4.1, n n V. ( 1 ), : 29

2 2.1.1 ( ) U V. U λ F, α, β U, α + β U λα U. 0 (, ),. ( 2 ): (1) : U V, W U, W V ; (2) ( ),, ;, U W U W., U W (?). V U W, U W?, α + β, α U, β W. U + W, U + W, U W, U W.. U 1, U 2,, U s V, {α 1 + α α s α j U j, 1 j s} (?), U 1, U 2,, U s, U 1 + U U s V, S V. V S S ( ), Span S, S Span S., S = S = {0}, Span S = 0 ; S = {α} α 0, Span{α} = {kα k F}, Fα; S = {α 1, α 2,, α s }, Span S = Span{α 1, α 2,, α s },,, Span{α 1, α 2,, α s } = {k 1 α 1 + k 2 α k s α s k j F, 1 j s}. Span S = {k 1 α 1 + k 2 α k s α s α j S, k j F, s 0}, Span S S ( )., U W U + W = Span U W. s j=1 U j V = M n (R) n. D = { }, U = { }, W = { }. D, U W V, U +W = V ; U W = D; U W ( ). D n; U W n(n + 1)/2; U + W = V n 2. (dim U + dim W ) dim (U + W ) = dim (U W ). (2.1.1) ( : A B = A + B A B, A A. ) 30

3 2.1.3 V = R[x] n, U = {f(x) V f(1) = 0}, W = {f(x) V f(2) = 0}. U + W = V, U W = {f(x) V f(1) = 0 f(2) = 0}. U W = {f(x) V f(1) = 0 f(2) = 0} (, x 1 + x 2 U W )., dim U = n 1 = dim W, dim (U W ) = n 2. (2.1.1) ( ) V, U W V. dim (U + W ) = (dim U + dim W ) dim (U W ). (2.1.2), 3. dim U = s, dim W = t, dim (U W ) = r. U W α 1, α 2,, α r. U W U W, U W U W, U W. α 1, α 2,, α r, β r+1, β r+2,, β s α 1, α 2,, α r, γ r+1, γ r+2,, γ t U W. α 1, α 2,, α r, β r+1, β r+2,, β s, γ r+1, γ r+2,, γ t U + W. (, U + W.), U +W, U W = 0, U W. U +W, U W. dim (U W ) = dim U +dim W R 2 x y ( 1 )., R 3 x, yoz ( 2 ) ( ) ( ). 0. ( ), ( ) n M n (F) {A M n A = λi, λ F} 0 {A M n tra = 0} ( ) U W V, : (1) U + W ( U W = 0); (2) α U + W, α = u + w, u U, w W, α = u + w, u = u, w = w ; (3) ; 0 = u + w, u U, w W, u = w = 0; (4) dim (U + W ) = dim U + dim W. : 2.1.3(3). 31

4 , (1) (4), (2) (3). (3)= (1)= (2). (3), α U W, α U W, 0 = α + ( α),, α = α = 0, U W = 0, (1). (1), α U +W α = u+w = u +w, u, u U, w, w W, u u = w w, u u U, w w W, u u = w w U W = 0, u = u, w = w, (2)., W 1, W 2,, W s W 1 W 2 W s (W 1 + W W s 1 ) W s. 0.,, ( ) W 1, W 2,, W s V, : (1) W 1 +W 2 + +W s dim (W 1 +W 2 + +W s ) = dim W 1 +dim W 2 + +dim W s ; (2) W j k j W k = 0, 1 j s, 1 k s; (3) α W 1 + W W s ; (4) R 3 x, y z, R 3 = R i R j R k α 1, α 2,, α n n V, V = Fα 1 Fα 2 Fα n n, V, U V. W V = U W ( U V, W ). W U. U (?) V = R 3. V 1 ; A 3, V = R 3. Ax = 0 U V 3 r(a). A ( A, ) Span{A 1, A 2, A 3 }. A m n : (1) A ( ) N(A): Ax = 0 ; (2) A (, )R(A): A ; 32

5 (3) A R(A T ): A ; (4) A ( ) N(A T ): y T A = 0 A T x = 0., N(A) R(A T ) F n ; R(A) N(A T ) F m, dim N(A) + dim R(A T ) = n; dim N(A T ) + dim R(A) = m.,.,. : A Hermite H A. Ax = 0 H A x = 0, N(A) = N(H A ),,, H A A, R(A) ;, H A A, A H A, R(A T ) = R(H T A ). H A A., A Hermite P ( ), P A = H A. r(a) = r, H A m r 0,, P m r y T A = 0, A N(A T ) A = , A. A Hermite, P : (A, I 3 ) = P A = H A, = = (H A, P ), A = 2, H A, A, R(A) = Span{A 1, A 2 } = Span{(1, 0, 1) T, (1, 1, 2) T }; R(A T ) = R(H T A ) = Span{(H1 A )T, (H 2 A )T } = Span{(1, 0, 1) T, (0, 1, 1) T }; N(A) = N(H A ) = Span{( 1, 1, 1) T }., N(A T ), P, A, xa = 0. N(A T ) = Span{(P 3 ) T } = Span{( 1, 1, 1) T }. 1.? 33

6 2.? 3.,? 4. R?? 5. C?? 6. 3 A. 7. F = C R. F n F ( 5 ).?? 8. F = C R. F n F ( 5 ). n?,, (T1) (T2). R, f R R (T1) (T2) ( ). (T2), x R, f(x) = xf(1), f(x) = kx,,. f (T1) (T2), R R 2, R, (!)f(x, y) = ax + by, a = f(1, 0) b = f(0, 1).., f R R 2? (T1) (T2)? θ, P (x, y) P (x, y )., : x = x cos θ + y sin θ, y = x sin θ + y cos θ. (2.2.1), P Q P Q, P + Q kp (!) P + Q kp! P (x, y) α, f(α), f(α + β) = f(α) + f(β); f(kα) = kf(α). (T1) (T2).,, R V ( ) : f(x) f (x), V., (f + g) = (f) + (g); (kf) = k (f)., f(x) x a f(x) d x V (T1) (T2),, n y (n) + f 1 (x)y (n 1) + f 2 (x)y (n 2) + + f n 1 (x)y + f n (x)y = 0, (2.2.2) y (i) = d i y, i = 1, 2,, n. L(y), L : y L(y) d x i n ( ), L (T1) (T2). (2.2.2) L {y : L(y) = 0}. 34

7 2.2.4 f(x) Laplace L(f(x)) = + f(t)e ixt d t. L (T1) (T2). (. ), (T1) (T2), U V. U V σ (T1) (T2), σ U V (linear transformation) (linear map). U ( ), U V Hom(U, V ) (, Hom F (U, V ))., Hom(V, V ) End V, Hom(V, F) V, V Hom R (R, R) = R,,,.,. σ Hom(U, V ). σ ( ), σ ( ).. σ Hom(U, V ), U V, U = V. 1. (T1) (T2) : σ(aα + bβ) = aσ(α) + bσ(β), a, b F, α, β U. (2.2.3) ( )., σ(a 1 α 1 + +a s α s ) = a 1 σ(α 1 )+ +a s σ(α s ), a j F, α j U. (2.2.4), f r- r x f(kx) = k r (x). f, r-, r = 1,. 2. F Q,. (T1), n, m f(2x) = f(x + x) = 2f(x), f(nx) = f(x + (n 1)x) = nf(x). f(nx) = f(m ( n m x)) = mf( n m x), f( n m x) = n m f(x), 35

8 r f(rx) = rf(x). f(0) = f(0 + 0) = f(0) + f(0), f(0) = 0, 0 = f(x + ( x)) = f(x) + f( x), f( x) = f(x), T(2)., F Q, (T2) (T1)., F C,, F = Q( 2 ), F 1. α = a + b 2 F, a, b Q, σ(α) = σ(a + b 2 ) = a, σ F, σ, σ( 2 2 ) = σ(2) = 2 2σ( 2 ) = ( ) ( ) ( ) r r cos θ θ r sin θ (2.2.5)., (1) σ(0) = 0; σ( α) = σ(α); (2) α 1, α 2,, α s, σ(α 1 ), σ(α 2 ),, σ(α s ) ; (3) σ(α 1 ), σ(α 2 ),, σ(α s ), α 1, α 2,, α s. ( (2) (3) 15.) σ(0) = 0, (?) ( ) α 1, α 2,, α n U, β 1, β 2,, β n V n, σ σ(α j ) = β j, 1 j n.. σ, σ U α = k 1 α 1 + k 2 α k n α n σ(α) = k 1 σ(α 1 ) + k 2 σ(α 2 ) + + k n σ(α n ) = k 1 β 1 + k 2 β k n β n., σ U V σ(α j ) = β j, 1 j n,. 1. σ,,,, , n,,,,. n n

9 2.2.8 ( ) R n R σ,,, σ, σ(e i ) = a i, 1 i n. x = (x 1, x 2,, x n ) T R n, n σ(x) = σ( x i e i ) = i=1 n x i σ(e i ) = i=1 n a i x i. i=1 R n R σ n., R n (R n ) n., (R n )., V V., F U V Hom F (U, V ) F. : (1) : V 0 ( V ),, 0 (, 0 V ); α V, 0(α) = 0. (2) : V,, I V ( I 1 ); α V, I(α) = α. V ( V ). (3) : k F. V α kα σ ( ) k, σ(α) = kα. k = 0 k = 1.. (4) : σ Hom(U, V ), τ Hom(V, U) α U, β V, τ(σ(α)) = α, σ(τ(β)) = β, σ, τ., σ, σ 1, (σ 1 ) 1 = σ( 16)., k(k 0), k 1., R 2, θ θ R ( ), R f f f A M n (F). F n A σ A : x Ax, σ A A, σ 1 : x A 1 x, (σ A ) 1 = σ A ( ) P n, X P 1 XP (2.2.6) M n (F), P. A A. 37

10 σ Hom(U, V ), {α U σ(α) = 0}, σ, Ker(σ) σ 1 (0);, {α V β U α = σ(β)}, σ, Im(σ) σ(u)., Kerσ Imσ U V ( 17), η(σ) r(σ), σ U = F n, V = F m, A m n. x U, U V σ σ(x) = Ax. σ A, σ n r(a) ( A ); σ A R(A), σ A r(a) U V. x U, U V ι ι(x) = x. ι,., U, V F ( ), σ Hom(U, V ). U Im(σ) σ σ(x) = σ(x). σ., σ., σ Ker(σ) ι U ι ( ). eσ Im(σ) ι V U, V F ( ), σ Hom(U, V ). (1) σ Ker(σ) = 0; (2) σ Im(σ) = V ; (3) σ σ., U = V, σ EndV, σ σ σ. (3), 16.,,., σ Hom(U, V ) τ : β V, σ, α U σ(α) = β, τ : β α. σ,. τσ = I U, στ = I V τ. σ Hom(U, V ),, σ.,? α U? α 1, α 2,, α n U, α 1, α 2,, α m V. σ(α 1 ) = a 11 α 1 + a 21α a m1α m, σ(α 2 ) = a 12 α 1 + a 22α a m2α m, σ(α n ) = a 1n α 1 + a 2nα a mnα m, 38

11 (σ(α 1 ), σ(α 2 ),, σ(α n )) = (α 1, α 2,, α m)a, (2.2.7) A = (a ij ) F m n σ α α., U = V, α α, σ α α σ α. (2.2.7) σ(α 1, α 2,, α n ), σ(α 1, α 2,, α n ) = (α 1, α 2,, α m)a. (,, ( )!!) (2.2.7),, ( ) σ Hom(U, V ) α α A, α U α x, σ(α) α A x σ e 1 = (1, 0) T, e 2 = (0, 1) T A, σ(e 1 ), σ(e 2 )., σ(e 1 ) = ( cos θ, sin θ) T, σ(e 2 ) = ( sin θ, cos θ) T. ( cos θ sin θ A = sin θ cos θ ) V = F n, σ V, e 1, e 2,, e n A. α = (x 1, x 2,, x n ) T = (e 1, e 2,, e n )(x 1, x 2,, x n ) T, σ(α) = σ(e 1, e 2,, e n )(x 1, x 2,, x n ) T = (e 1, e 2,, e n )A(x 1, x 2,, x n ) T = A(x 1, x 2,, x n ) T. σ : α Aα., F n ( )., A = (a ij ), σ(α) = A(x 1, x 2,, x n ) T = (a 11 x 1 + a 12 x a 1n x n,, a n1 x 1 + a n1 x a nn x n ) T = (σ 1 (α),, σ n (α)), σ j (α) = a j1 x 1 + a j2 x a jn x n, 1 j n n ;,.,,..,?, { α 1, α 2,, α n }, {β 1, β 2,, β n } 39

12 U, P α- β- ; { α 1, α 2,, α m}, { β 1, β 2,, β m} V, Q α β -. σ Hom(U, V ) α α - A, β β B. α = (α 1, α 2,, α n )x, σ(α) β?, α β P 1 x; (2.2.7), σ(α) α Ax, β BP 1 x., Q 1 Ax = BP 1 x. Q 1 A = BP 1, QB = AP B = Q 1 AP. (2.2.8), α- α - σ α-α -, σ α- i α - i. (2.2.8) ( ) (α )U σ A V (β ) P 1 (α )U B σ Q 1 V (β ) (2.2.8) α-α - β-β - ;, A B, (2.2.8), σ, A B σ α-α - β-β -, 5. U = V α- β-, α - β -, P Q, (2.2.8) B = P 1 AP. (2.2.9) V n, σ V. α 1,, α n β 1,, β n V, A B σ. A B. B = P 1 AP,, P.., σ σ, σ tr(σ) σ R 2 π/4. σ ] A = [

13 σ (1, 0) T, (1, 1) T [ 0 2 B = ]. (1, 0) T, (1, 1) T T = [ AT = T B, B = T 1 AT. σ = 1, tr(σ) = 2. ].? (2.2.9),., A B, (2.2.9), σ EndV σ α- A; V α -, α- α - P, σ α - B!!, σ τ,? 18. (,.) σ EndR 2 x- : σ((x, y) T ) = (x, 0) T. σ (α- ) A = ( ) σ (α - ){β 1 = (1, 1) T, β 2 = (1, 1) T } B = 1 2 ( ) (2.2.9), B = P 1 AP, P α- α - P = ( ) ( α -, P (β 1, β 2 ).), A B A 2 = A, B 2 = B.., σ ( ) F U V dim F U = dim F V.., U V, U V., U V, U V

14 2.2.5,, F n V F n,,, V F n,, U = F[x] n, V = F n (, F n ). n, U V., U V U. U V σ: n 1 σ : f(x) = a i x i (a 0, a 1,, a n 1 ). σ, U f(x) = n 1 i=0 i=0 a i x i ( ) (a 0, a 1,, a n 1 ). U, U (, 1 + x 2 x + x 2 (1, 1, 0, 0) (2, 1, 1, 0,, 0), 3 + x 2, (3, 0, 1, 0,, 0)., 3 + x 2 )., U V! ( n,,,,.) A, A = n. F A = F n ; F A ( ) F n ( 24) V = { }?, V R, dim V 1. x log x V R, dim V = dim R = 1.,,, (, x log x).. Hom(U, V ). Hom(U, V ),. σ, τ Hom (U, V ), α V, k F, σ + τ : α σ(α) + τ(α); kσ : α kσ(α). (2.2.10), (2.2.10) σ + τ kσ U V, σ τ k σ., Hom (U, V )., Hom (U, V ), F. dim F U = n, dim F V = m, ( 21) dim F Hom(U, V ) = mn = (dim F U)(dim F V ). (2.2.11), U = V, End V σ τ στ : α σ(τ(α)), α V (2.2.12) 42

15 ,, (2.2.12),. σ σσ σ 2., k, σ k σ k = σ k 1 σ (, σ 0 = I)., F f(x), σ f(σ). ( ), σ, σ f(σ) = f(σ)σ., σ 2 = σ. σ k = 0 ( σ ) V = F n. α = (x 1, x 2,, x n ) T, σ(α) = (x 1, 0,, 0) T ; τ(α) = (x 2, x 3,, x n, 0) T. σ τ V, σ, τ (?) , ;, V σ, V = Im(σ) Ker(σ) (!), σ V ( Ker(σ)) Im(σ) ( ) V F n, α 1, α 2,, α n V. M n (F) F n. σ End V, A(σ) σ. End V M n (F) ψ ψ : End V M n (F) σ A(σ) ψ (, ), : (1) A(σ + τ) = A(σ) + A(τ); (2) A(kσ) = ka(σ), k F; (3) A(στ) = A(σ)A(τ); (4) σ A(σ) ; A(σ) 1 = A(σ 1 ); (5) A(0) = 0; A(I) = I. ψ, 22. ψ. ψ, Ker(ψ) = 0. ψ(σ) = 0, A(σ) = 0. α V, x α, σ(α) A(σ)x = 0x = 0. σ = 0. ψ. B n. σ(α) = (α 1, α 2,, α n )Bx. σ V, σ B, ψ(σ) = B, ψ σ EndR 2. α = (x, y) T R 2, σ(α) = σ(xe 1 ) + σ(ye 2 ) = xσ(e 1 ) + yσ(e 2 ). 43

16 σ(e 1 ) = (a 11, a 12 ) T, σ(e 2 ) = (a 21, a 22 ) T. ( a11 a σ(α) = xσ(e 1 ) + yσ(e 2 ) = 12 a 21 a 22 ) ( x y ). σ. ( a11 a 12 a 21 a 22 ) (, ) (F) (F).,. End V V ψ End V M n (F), End V M n (F).,! End V,,,,. V n, End V, Aut V, V, ( ) , Aut V n GL n (F),, (F ),.,, Aut V GL n (F) ( 23): ( ) U V n m F. Hom F (U, V ) = F m n., V V. V V, V = R n. V = (R n ) n.. R n ( )v = (v 1, v 2,, v n ), (v) : x = (x 1, x 2,, x n ) R n, (v) : (x 1, x 2,, x n ) v T x = v 1 x 1 + v 2 x v n x n (v) v, R n (R n ) R n v (R n ) ( 1, v) v. : R ( 1, v) v (R n ) R n v

17 , ( )., R 2 (R 2 ) ( )(A, B) (R 2 ) z = Ax + By, R 3 ( 1, A, B) ( ) ( ) n V α 1, α 2,, α n, V α1, α 2,, α n : α i (α j ) = δ ij, 1 i, j n. (2.2.13) α 1, α 2,, α n., R 2 e 1, e 2 e 1, e 2, e 1(x, y) = x, e 2(x, y) = y , f(x, y) = a 1 x + a 2 y (R 2 ) ( R 2 ), f = a 1 e 1 + a 2e ( ) V V (V ) V, V. ( )., V, V = V = V. V V,. v V, f V, V V φ : φ : v φ(v) : f f(v). φ V V, ? 2. ( R 2 )?? ( R 3 )?? 3.? 4. (2.2.8)? V 1, EndV AutV?, EndR, AutR, EndC, AutC? 7.,? 8. R = R R 3 = (R 3 )? 9. V x + y + z = 0, V?,,,. V n, U V. W = {α V (α, β) = 0, β U}. ( 30), W V, U, U V n, U V. V = U U., dim U = n dim U. 45

18 , V U, U : U ( )α 1, α 2,, α s V α 1, α 2,, α s, α s+1,, α n, U = Span{α s+1, α s+2,, α n }., U = W, W = U, : (U ) = U., V., v V v, dim v = dim V 1, v V. m n A : N(A) R(A ) F n ; R(A) N(A ) F m. R(A ) ; R(A) N(A ) ;, ( )., N(A) (1) N(A) = R(A ) ; (2) R(A) = N(A ). A F m n. (1), (2), 31. x N(A), Ax = 0, A (i) x = 0, i = 1, 2,, m. C n ( ), Ā( : A!) A (i) x. Ā x. x R(A ) (Fourier ) V = {f(x) f(x) = m (a n cos nx + b n sin nx), a n, b n R}. V 2m + 1. : (f(x), g(x)) = 1 π π π n=0 f(x)g(x) d x. 1 2, cos x, sin x, cos 2x, sin 2x,, cos mx, sin mx V ( 1 π ). c ( 34) c = 1 = Span{ cos x, sin x, cos 2x, sin 2x,, cos mx, sin mx}. a n = 1 π π π f(x) cos nx d x, b n = 1 π π π f(x) sin nx d x., a n = (f(x), cos nx) b n = (f(x), sin nx) f(x)., : ( ) U V, β V. α U β α β γ, γ U. α β U ( ), ˆβ( U). 46

19 , β U ( ) U β ( ), (?) ( ) U R 3 U,, α = (x, y, z) U α U, ˆβ = Proj U α R 3, α = (x, y, z) x- (x, 0, 0) ( ) U V, β V, α U. α β U β α U. γ U, β γ = (β α) + (α γ), β α U, α γ U, β γ 2 = β α 2 + α γ 2 β α 2., γ U, β α β γ, α β U., β U, β V = U U, β = α + γ, α U, γ U, α β U ˆβ.,? U = Rv 1. β U ˆβ = xv,, (β xv, v) = 0, x = (β, v) (v, v) (2.3.1) ˆβ = (β,v) (v,v) v, β v Proj vβ, γ = β Proj v β β U ( U ). U = Rv 1 Rv 2 2, ˆβ = x 1 v 1 + x 2 v 2 (2.3.2) (β ˆβ, v i ) = 0, i = 1, 2, (β x 1 v 1 x 2 v 2, v i ) = 0, i = 1, 2 (2.3.3) v 1, v 2, (2.3.3), ˆβ., v 1, v 2, x 1 = (β, v 1) (v 1, v 1 ), x 2 = (β, v 2) (2.3.4) (v 2, v 2 ) ˆβ (2.3.2) x 1 v 1 x 2 v 2 β v 1 v 2 Proj v1 β Proj v2 β. β U β U Proj v1 β + Proj v2 β., U 0, ( 35) 47

20 2.3.1 β V, U V, α 1,, α s U, β U ˆβ = Proj v1 β + + Proj vs β = (β, α 1) (α 1, α 1 ) α (β, α s) (α s, α s ) α s. (2.3.5), α 1,, α s U, β U ˆβ = (β, α 1 )α 1 + (β, α 2 )α (β, α s )α s. (2.3.6) β U Proj U β, Proj U β = ˆβ β U U, U R 3, α = (x, y, z) xoy (x, y, 0), α xoy ( ){(1, 0, 0), (0, 1, 0)} (x, 0, 0) (0, y, 0).. Ax = b, x = (x 1, x 2,, x n ) T R n Ax b., x 0 R n x R n, Ax 0 b Ax b, x 0. ( : y = Ax b, b ( )R(A). y = Ax b y = Ax b 2 = (Ax b) T (Ax b) = x T A T Ax 2x T Ab + b T b,,,.) x R n, Ax A R(A). Ax = b x 0 Ax 0 b R(A)., Ax 0 b R(A) = N(A T )., Ax 0 b A T (Ax 0 b) = 0, x 0 A T Ax = A T b. (2.3.7) Ax = b. 5, (2.3.7)!,, Ax = b,! A T Ax = A T b Ax = Proj R(A) b, Ax = b Ax = Proj R(A) b. Proj R(A) b b x 1, x 2. : 48

21 : b = a 1 x 1 + a 2 x 2. b x 1 x , a 1 6a 2 = 1 a 1 2a 2 = 2 a 1 + a 2 = 1 a 1 +7a 2 = 6 Ay = β, A = , β = , y = ( a1 a 2 ),,.. A A 1, A 2, (2.3.5) β R(A) ˆβ = (β, A 1) (A 1, A 1 ) A 1 + (β, A 2) (A 2, A 2 ) A 2 = 8 4 A A Ay = β (?) b = 2x x 2. a 1 = 2, a 2 = 1/2.. V, (α, v) = 0, v V, α = 0 V Minkowski 18, Minkowski. : (α, β) = 0, α, β V. Minkowski,., Minkowski ( ). 4 Minkowski V ( Minkowski 4- ), e 0, e 1, e 2, e 3, (e 0 ) 2 = (e 1 ) 2 = (e 2 ) 2 = (e 3 ) 2 = 1., α = (x 0, x 1, x 2, x 3 ) T β = (y 0, y 1, y 2, y 3 ) T (α, β) = x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3, 18 Hermann Minkowski( ),, (geometry of numbers),, Albert Einstein ( ). 49

22 α = (x 0, x 1, x 2, x 3 ) T, : (1) ( spacelike): α 2 > 0; (2) ( timelike): α 2 < 0; (3) ( lightlike): α 2 = 0. α 2 = x x x x 2 3. ( ) (light cone)., V W : (a) ( spacelike): α 2 > 0, α W ; (b) ( lightlike): α 2 0, α W 0 α W α 2 = 0; (c) ( timelike): ( ) W Minkowski 4- V. (1) W W, ; (2) W W W 0 W. : ( = ) ( ) ? 3.? 4. R 3 (x, y) = x 1 y 1 + x 2 y 2 x 3 y 3,? x?,,??., ( 38 ) V, σ End V. σ, α, β V, d(σ(α), σ(β)) = d(α, β), σ.. ; R 2 ; V, σ End V. σ σ σ. 50

23 α V, α = (α, α) = d(α, 0),.,. σ, (σ(α + β), σ(α + β)) = (σ(α), σ(α)) + (σ(α), σ(β)) + (σ(β), σ(α)) + (σ(β), σ(β)) = (α, α) + (σ(α), σ(β)) + (σ(β), σ(α)) + (β, β). (α + β, α + β) = (α, α) + (α, β) + (β, α) + (β, β) = (α, α) + (α, β) + (β, α) + (β, β). (σ(α), σ(β)) + (σ(α), σ(β)) = (α, β) + (α, β), (σ(α), σ(β)) (α, β). (!) Re(α, iβ) = Im(α, β), (α, iβ) (α, β), (σ(α), σ(β)) (α, β), (σ(α), σ(β)) = (α, β), V n, α 1, α 2,, α n V, σ End V, A σ. σ A. α V, α = (α 1, α 2,, α n )x, x C n, α 1, α 2,, α n, α = (α, α) = x x. σ(α) = β = (α 1, α 2,, α n )Ax, β = (Ax) (Ax) = x (A A)x. A A = I, σ(α) = β = x x = α, σ, σ., σ, σ, x C n ( R n ), (β, β) = x (A A)x = (α, α) = x x. Hermite x (I A A)x = 0 x, I A A = 0( 46), A A = I. A., 2.4.2,, σ π/2, {e 1, e 1 + e 2 } ( ) , ;.,. Aut V,.,., GL n (R), V, σ End V. σ σ. 51

24 α 1, α 2,, α n V, A σ, β j = σ(α j ), 1 j n. σ, 2.4.1, σ, β 1, β 2,, β n., σ, σ, 4,, σ ( ) R ± I, ( x x). R 2 : e 1, e 2. σ A, A, ( ) c s Q = (2.4.1) s c ( ) c s P = s c (2.4.2) c 2 + s 2 = 1. Q, P, ( y = 1 c s x, 39), ( ) R 3, 6 ( 40): , 1, cos θ sin θ, sin θ, cos θ cos θ sin θ, sin θ. cos θ,., 6., :, A = ,. A 1, α = (x 1, x 2, x 3 ) T, Aα = α α = ( 3, 1, 1) T, (0, 0, 0) ( 3, 1, 1). A,, B = cos θ sin θ 0 sin θ cos θ 52 = Q 1 AQ,

25 tr B = tr A, cos θ = = 3 4, cos θ = 7 8..,,,,.., x x, y = x 2 x., Householder 19 Givens 20, ( Householder ) v C n, n H : H = I 2vv v v. (2.4.3) H Householder. H H v Householder, x C n H v x = x 2vv v x. (2.4.4) v, Householder H v v ( v, v v, 43). x C n = Span{v} v x = P v x + P v x,. H v x = P v x P v x. v H v x Pv x x P v x P v x Cv v = e 3 R 3, v v = 1, Householder H v H v x = x 2vv x. H v xoy ( Givens ) (2.4.1) Jacobi 21 ( 2 ) Givens., c 2 + s 2 = 1, θ = arctan s c, n G(i, j, θ) = I n (1 c)(e ii + E jj ) + s(e ij E ji ) (2.4.5) 19 Alston Scott Householder( ),, Householder., Association for Computing Machinery. 20 James Wallace Givens( ),,.. 21 Carl Gustav Jacob Jacobi( ),, Jacobian Conjecture ( ),. 53

26 Givens ( ). ( 44). Householder Givens 0,.,, σ V. α, β V, (σ(α), β) = (α, σ(β)) (2.4.6) σ...,, σ σ. σ α 1, α 2,, α n A. α, β x, y. σ(α) = (α 1, α 2,, α n )Ax, σ(β) = (α 1, α 2,, α n )Ay, (σ(α), β) = (Ax, y) = y T Ax = (A T y) T x = (Ay) T x = (x, Ay) = (α, σ(β)).. σ, α 1, α 2,, α n A = (a ij ). σ(α j ) = n i=1 a ijα i. a ij = (σ(α j ), α i ) = (α i, σ(a j )) = (σ(α i ), α j ) = a ji,, σ ,,.,,, (x, y) T (x + y, x) T, 0 x, y 1 (1,1) (2,1) ,,. 54

27 2.4.3 σ, τ V. α, β V, (σ(α), β) = (α, τ(β)) (2.4.7) τ σ. ( 45 ), σ, σ. σ = σ, σ.,., σ ( 45 )σ σ σ : (x, y) T (a 1 x + b 1 y, a 2 x + b 2 y) T. σ : (x, y) T (a 1 x + a 2 y, b 1 x + b 2 y) T. σ σ!,.,, σ, τ V, A, B σ, τ. (1) (σ ) = σ; (2) (σ + τ) = σ + τ ; (3) (λσ) = λσ, λ R; (4) (στ) = τ σ ; (5) τ = σ B = A T ( ) 2.3.1, x V x = P U x + P U x, P U x U, PU x U x U., P U PU V U U, P U : x P U x P U : x P U x. P U = P U P U + P U = I., U = Span{v} 1, U v, P U x P v x σ V. σ V U σ. 55

28 . σ V U, σ = P U. σ. x, y V, (P U x, y) = (P U x, P U y + P U y) = (P U x, P U y) = (x P U x, P U y) = (x, P U y), σ.. σ σ V Im(σ). σ, Im(σ) = Ker(σ), σ A M n (C) x Ax A 2 = A, A = A., A C n A R(A). 1. ( R 2 ),? 2. ( R 3 )? 1 1?,?? 3.?? ,,? 5.? 6.?? ? :,. F U V.,,. U V ( )U V = {(u, v) u U, v V }; U V, (u, v), (u 1, v 1 ), (u 2, v 2 ) U V λ : (u 1, v 1 ) + (u 2, v 2 ) = (u 1 + u 2, v 1 + v 2 ), λ(u, v) = (λu, λv)., U V, U V, U V, U V. ( 47): ( ) α 1, α 2,, α n ( α- ) β 1, β 2,, β m ( β- ) U V, U V, α β-., (α i, 0), (0, β j ), 1 i n, 1 j m (2.5.1) dim (U V ) = dim U + dim V. (2.5.2) ( U V, U V, ((u 1, v 1 ), (u 2, v 2 )) = (u 1, u 2 ) + (v 1, v 2 ). (2.5.3) 56

29 48.),.. U V U V W = U V {(u, v) W u U, v = 0} {(u, v) W v V, u = 0}. R 2 x y,, x y. U V (u, v) u + v,., α β- α β-... σ i Hom(U i, V i ), i = 1, 2. U 1 U 2 V 1 V 2 σ ( u i U i ): σ(u 1 + u 2 ) = σ 1 (u 1 ) + σ 2 (u 2 ) (2.5.4) ( 49), σ, σ 1 σ 2, σ 1 σ 2. : σ 1 σ 2 α-α - β-β - A B, σ 1 σ 2 α β-α β - A B. σ 1 σ 2 α β-, σ 1 σ 2 (α i ) = σ 1 (α i ), σ 1 σ 2 (β j ) = σ 2 (β j ). σ 1 σ 2 (α β) = (σ 1 (α 1 ),, σ 1 (α n ), σ 2 (β 1 ),, σ 2 (β m )) = (α β)(a B) σ, τ (R 2 ) σ(x, y) = ax + by τ(x, y) = cx + dy. σ τ Hom(R 2 R 2, R R) : σ τ(x, y, u, v) = (ax + by, cu + dv)., σ τ - C = ( a b 0 ) c d σ τ A = (a b) B = (c d), C = A B , ( 50): σ i Hom(U i, V i ), 1 i n. σ α i -α i - A i. n σ i Hom( n n U i, V i ) n α i - n α i - n A i. i=1 i=1 i=1 i=1 57 i=1 i=1

30 .,. V = U W, σ EndV. σ σ 1 EndU σ 2 EndW, σ = σ 1 σ 2, σ(u + w) = σ 1 (u) + σ 2 (w)., u U, w W, σ(u) = σ 1 (u) U, σ(w) = σ 2 (w) W., σ(u) U, σ(w ) W. U( W ) σ.., V = V 1 V 2 V s, σ EndV. V i σ, σ i EndV i, 1 i s, σ = σ 1 σ 2 σ s. V i σ, V i σ i : σ i, v V, σ i (x) = σ(x), x V i. v = v 1 + v v s, v i V i, σ 1 σ 2 σ s (v) = σ 1 (v 1 ) + σ 2 (v 2 ) + + σ s (v s ) = σ(v 1 ) + σ(v 2 ) + + σ(v s ) = σ(v 1 + v v s ) = σ(v), σ = σ 1 σ 2 σ s. σ i σ V i, σ Vi V = V 1 V 2 V s, σ EndV. V i σ, V, σ. σ 2.5.4, σ = σ 1 σ 2 σ s. σ i V i α i - A i, s i= σ α i - s A i. i=1 σ EndV. v, x P v x. V = Ker(σ) Im(σ) (2.5.5) v = (v σ(v))+σ(v), v σ(v) Ker(σ), σ(v) Im(σ) Ker(σ) Im(σ) = 0. Ker(σ) Im(σ), V,, σ 0 n r I r, 58

31 r σ, n V., σ Ker(σ) 0, Im(σ)., A x Ax,.,,. U V, (α i, β j ), 1 i n, 1 j m (2.5.6), U V 22, U V., U V (2.5.6) α i β j, 1 i n, 1 j m (2.5.7) U V n m x ij α i β j. (2.5.8) i=1 j= U V, dim (U V ) = (dim U)(dim V ).., ( (2.5.8) ) U = F 1 n V = F m 1. U V e T 1,, et n e (m) 1,, e (m) m ( e (m) i i m ). F 1 n F m 1 : e T i e (m) j, 1 i n, 1 j m. (2.5.9) e T i e (m) j E ji, F 1 n F m 1 = F m n. (2.5.10) ( ) F[x] F[y] = F[x, y]. F[x] F[y] 1, x, x 2,, x n, 1, y, y 2,, y m, 22 ( ) Gregorio Ricci-Curbastro ( ), Tullio Levi-Civita ( ) ( ). 59

32 F[x] F[y] x n y m, m, n 0. x n y m x n y m.,,.. σ i Hom(U i, V i ), i = 1, 2., U 1 U 2 V 1 V 2 σ 1 σ 2 ( u i U i ): σ 1 σ 2 (u 1 u 2 ) = σ 1 (u 1 ) σ 2 (u 2 ) (2.5.11) ( 51), σ 1 σ 2, σ 1 σ 2., U 1 U 2 (2.5.8), (2.5.11) U 1 U 2,, , σ (F 2 ), τ Hom(F, F 2 ). σ τ - A = (a b) B = (x y) T. σ τ F 2 F σ τ(e 1 1) = σ(e 1 ) τ(1) = a (x, y) T = ax(1 e 1 ) + ay(1 e 2 ) σ τ(e 2 1) = σ(e 2 ) τ(1) = b (x, y) T = bx(1 e 1 ) + by(1 e 2 ). ( ax bx σ τ {e 1 1, e 2 1}-{1 e 1, 1 e 2 }- ay by A = (a b) B = (x y) T., ) ( ) A = (a ij ) B = (b st ) m n p q. A B ( ) ( ) Kronecker, A B, mp nq a 11 B a 12 B a 1n B a 21 B a 22 B a 2n B. a m1 B a m2 B a mn B (1) e (n) i (2) I n I m = I mn ; ( ) a (3) ( x y ) = b (4) ( a b ) ( ) x = y (e (m) j ) T = E ij ; ( ) ax ay = bx by ( ) ax bx ( a ay by ( ) a (x ) y. b b ) ( ) x = ax + by. y A, B 2. σ F 2 2, σ : X σ(x) = AXB T., σ ( ) E 11, E 12, E 21, E 22 A B, ( ) E 11, E 21, E 12, E 22 B A. 60

33 AXB = C. ( 52) (1)( ) (A B) C = A (B C); (2)( 1) (A B)(C D) = AC BD; (3)( 2) A (B ± C) = A B ± A C; (4)( ) (A B) T = A T B T ; (5)( ) (A B) 1 = A 1 B 1 ( A B A B ); (6) A B m n, A B = A m B n ; (7)( ) r(a B) = r(a)r(b); tr(a B) = tr(a)tr(b); (8)( ) λ µ m A n B, x y, λµ A B, λ + µ A I m + I n B, x y , ( 53): σ i Hom(U i, V i ), i = 1, 2. σ i α i -α i A i, i = 1, 2. σ 1 σ 2 α 1 α 2 -α 1 α 2 A 1 A 2. (2.5.7),,. U V. α V, V {β V β = α + u, u U} α U, α + U. α 1 α 2,, α 1 + U = α 2 + U. V U V/U. ( 54), (α 1 + U) + (α 2 + U) = (α 1 + α 2 ) + U, a(α + U) = aα + U. (2.5.12) V/U V U., V/U α + U ᾱ V = R 2 U = x-. α R 2, α + U x- α x-, x- x-, V/U ( ). V/U x-, y-, V/U y-, V/U = y , V U α + U U α,,,.,.,., Z, 2Z. a b 2Z, a b., n n + 2Z = 0 + 2Z = 0;, m m + 2Z = 1 + 2Z = 1. 61

34 Z/2Z 2 { 0, 1}. + ( Z ) : = 0, = = = 0 1 = 1 0 = 0, 1 1 = 1 (2.5.13), (2.5.13) Z/2Z + =, =., Z/2Z Z 2 F 2., , F 2 = {0, 1}. p, 2Z pz, F p = Z/pZ.,, (1) u U, u + U = 0 + U = U, U V/U 0 ; (2) v + U = 0 v U; (3) α + U = β + U α β U V = U W, V/U = W., dim V/U = dim V dim U. α 1,, α s U, α s+1,, α n W. V. α 1,, α s, α s+1,, α n V/U., : α s+1 + U,, α n + U (2.5.14) k 1 (α s+1 + U) + + k n s (α n + U) = 0, (k 1 α s k n s α n ) + U = 0, k 1 α s k n s α n U, 0., α + U V/U, α = x 1 α x s α s + x s+1 α s x n α n, α + U = (x 1 α x s α s + x s+1 α s x n α n ) + U = [(x 1 α x s α s ) + U] + [(x s+1 α s x n α n ) + U] = (x s+1 α s x n α n ) + U = x s+1 (α s+1 + U) + + x n (α n + U), (2.5.14) V/U., α 1,, α s U, α 1,, α s, α s+1,, α n V. α s+1 + U,, α n + U V/U. 62

35 2.5.2 σ : V, σ EndV. U σ. V/U σ : α + U σ(α) + U. (2.5.15) σ ( 56), σ σ R 3 xoy-. σ e 3, e 1, e 2 (0) I 2. U z- ( R 3 ), σ V/U σ((x, y, z) T + U) = (x, y, 0) T + U = (x, y, z) T + U, σ ē 1, ē 2 I 2. σ e 3, e 1 + e 3, e 2 + e σ e 1 + e 3, e 2 + e 3 I 2 ( σ ) A M s (F), B, C M n s (F). V, σ EndV. σ V α 1,, α s, α s+1,, α n α 1,, α s, β s+1,, β n A B A C. U = Span{α 1,, α s } σ, σ σ V/U ᾱ s+1,, ᾱ n β s+1,, β n B C. σ, U = Span{α 1,, α s } σ, Span{α s+1,, α n } Span{β s+1,, β n } σ , ᾱ s+1,, ᾱ n β s+1,, β n V/U. σ ᾱ 1 = 0,, ᾱ s = 0, ( σ(ᾱ s+1 ),, σ(ᾱ n )) = (ᾱ 1,, ᾱ s, ᾱ s+1,, ᾱ n ) ( σ(ᾱ s+1 ),, σ(ᾱ n )) = (ᾱ s+1,, ᾱ n )B, ( ) 0. B σ ᾱ s+1,, ᾱ n B. σ β s+1,, β n C. Span{α s+1,, α n } Span{β s+1,, β n } σ, 57., A, B, C, A B A C B C.. U V, U V F (U V ), U V, F (U V ) = { n x i (u i, v i ) u U, v V, x i F, n N} (2.5.16) i=0 63

36 F (U V ) N: (u 1 + u 2, v) (u 1, v) (u 2, v) (u, v 1 + v 2 ) (u, v 1 ) (u, v 2 ) x(u, v) (xu, v), x(u, v) (u, xv) F (U V )/N U V ( ). F (U V )/N U V,, F (U V ), N , U = V = F. U V α β N : (x 1 α + x 2 α, yβ) (x 1 α, yβ) (x 2 α, yβ) (xα, y 1 β + y 2 β) (xα, y 1 β) (xα, y 2 β) z(xα, yβ) (zxα, yβ), z(xα, yβ) (xα, zyβ) F (U V )/N, (xα, yβ) + N = xy(α, β) + N, (α, β) + N F (U V )/N. F (U V )/N 1., (α, β) α β F (U V )/N U V, 61., ( 62) U V, F (U V )/N U V..,. P 1. U i, V j, 1 i n, 1 j m. Hom( n mp U i, V i),. 2. Q( 2) 2. Q ( )? 3. C R 2. C/R? 4. p, F p = Z/pZ? F 2( F p) ( )? 5. R[x] C R,? 6. V, σ EndV (,, ). U σ, σ σ V/U, σ (,, )? i=1 j=1 :,,,,,.. 64

37 ,,,, (x 1, y 1 ),, (x n, y n ), y = a 0 + a 1 x + a 2 x 2., Y = Xα X, α ( ), y. ( )Y Xα, Y = Xα, X T Y = X T Xα ( 2, 12), ( 1, 5), (0, 3), (1, 2), (2, 4).( 63) S = {{x k } = (, x 2, x 1, x 0, x 1, x 2, ) x i R}, S, V. S ( ). S, l 2 = {{x k } S S : (1) (2) + k= l 1 = {{x k } S x k = 0, k 0 x 2 k < + }. + k=1 x k < + }; {{x k } S x k 0 } ( x k 0 ) (3) l = {{x k } S x k = 0, k 0, C, x k < C, k}. l 2, ( 64), ({x k }, {y k } ) = 65 + k= x k y k. (2.6.1)

38 {x k } {y k } + k= x k y k = 0.,,,, {x k } {y k }, x k y k = 0, k ( {x k } {y k } 0, ), {x k } {y k }....., n (a i a 0 a n 0) y k = a n x k + a n 1 x k a 1 x k+n 1 + a 0 x k+n (2.6.2) {x k } {y k }. {y k },, ( ) y k = x k x k x k+2 ( x = cos πt 4 ) {x k } = {, x 0 = 1, ( 65) {y k } = {, y 0 = 1 2, y 1 = 1, 1, 0, 1, 1, 1 1, 0,, 1, 1, 0, } , 0, 1, 1, 1 1, 0,, 1, 1, 0, } {u k }( x = cos 3πt 4 ), ( 65) {u k } = {, u 0 = 1, 1 2, 0, 1 1, 1,, 0, 1, 1, 1, 0, } , ( ) n n.,. n,, : y k+1 = Ay k, k (2.6.3) y k R n, A M n (R). ( a 0 = 1) a n x k + a n 1 x k a 1 x k+n 1 + x k+n = 0, k y k+1 = Ay k ( k), y k = (x k, x k+1,, x k+n 1 ) T, A = a n a n 1 a n 2 a 1 66

39 , A f(x) = a n + a n 1 x + + a 1 x n 1 + x n. x k = 0, k < 0, (2.6.3) y k = A k y 0 (2.6.4) y 0 = (x 0, x 1,, x n 1 ) T. A..., 2 3 ( ).,., S V ( R 2 R 3, ), σ V, σ(s) = {y V y = σ(x), x S} S. ( 67) V S vol(s). vol(σ(s)) = σ vol(s) , S = {(x, y) x2 + y2 1}. a 2 b 2. ( ) a 0 α Aα, A = S A π = πab. 0 b ( AB = A B ) A, B M n (F), G F n V 0. F n σ : x Ax τ : x Bx στ : x (AB)x , τ(g) B V, σ(τ(g)) A B V, στ(g) AB V. σ(τ(g)) = στ(g), AB V = A B V.. AB = A B. A 1 XB 1 + A 2 XB A s XB s = C (2.6.5), A i M m (C), B i M n (C), C C m n, X C m n. (2.6.5)., AXB T = C C σ, σ : X AXB T C m n , σ ( ) E ij B A, X ( ) x, σ σ : (E 11,, E mn )x (E 11,, E mn )(B A)x (2.6.6) AXB T = C (B A)x = c (2.6.7) c C ( )., m n ( ) mn, 67

40 2.6.1 A = (a ij ) m n, A mn, A, vec(a), vec(a) = (a 11, a 21,..., a m1, a 12, a 22,..., a m2,..., a 1n, a 2n,..., a mn ) T., A : rvec(a) = (a 11, a 12,..., a 1n, a 21, a 22,..., a 2n,..., a m1, a m2,..., a mn )., vec F m n F mn, ( )., rvec(a T ) = (vec(a)) T, vec(a T ) = rvec(a) T. ( ), (1) rvec(abc) = rvec(b)(a T C), vec(abc) = (C T A)vec(B)., vec(a m m B m n ) = (I n A)vec(B), vec(b m n C n n ) = (C T I m )vec(b); (2) vec(xa + yb) = xvec(a) + yvec(b), x, y C; (3) vec(ac + CB) = (I n A + B T I m )vec(c). (2.6.5) X C m n (2.6.5) vec(x) Gx = c, x = vec(x), c = vec(c), G = s i=1 (BT i A i ). (2.6.5), c = vec(c) = s i=1 vec(a ixb i ) = s i=1 (BT i A i )vec(x) = G vec(x) = Gx (2.6.5) r([g, c]) = r(g); G Lyapunov 23 AX + XB = C, A = ( ) ( 2 1, B = 0 2 ) ( 2 3, C = 3 4 ). G = I 2 A + B T I 2 = , c = vec(c) = ( 2, 3, 3, 4) T. Gx = c x = ( 1, 1, 1, 1) T. 23 Aleksandr Mikhailovich Lyapunov( ),,,,.. 68

41 X = ( ) (1) ( 2.1.2); (2) V. : dim V = max{m 0 = V 0 V 1 V m 1 V m = V, V i V i+1 } ( ) A A, A. 6. V, W 1, W 2,, W s V. W 1 W 2 W s V.( : Vandermonde.) 7. V n, U V. U V, U. 8. V n, U = {f(x) V f(1) = 0}. U V, V.. 9. U = [(1, 2, 3, 6) T, (4, 1, 3, 6) T, (5, 1, 6, 12) T ], W = [(1, 1, 1, 1) T, (2, 1, 4, 5) T ] R 4, (1) U W ; (2) U W, U ; (3) U W, W ; (4) U + W. 10. U = {(x, y, z, w) x + y + z + w = 0}, W = {(x, y, z, w) x y + z w = 0}. U W, U + W 11. A, B n m, m p, V xab = 0. U = {y = xa x V } F n, U. 12. A n. (1) A. ; (2) A Hermite Hermite. ; (3) R ; (4) (1)-(3). 13. F : f(x), g(x), g(x) 0, q(x) r(x) f(x) = g(x)q(x) + r(x) r(x) = 0 r(x) < g(x) (1) f. : f,, ; (2) (1). 69

42 15. σ(α 1 ), σ(α 2 ),, σ(α s ), α 1, α 2,, α s. 16.(1) 2.2.2; (2) σ Hom(U, V ),, τ = σ 1, σ = τ 1 ; (3) 2 C. 17. σ Hom(U, V ). (1) σ Kerσ Imσ U V ; (2) U/Kerσ Imσ. 18. σ EndV V A. : τ EndV, τ σ, τ A ? R n. 21. U V, dim F Hom(U, V ) = (dim F U)(dim F V ) ( 2.2.7) φ V V. 25. : f(x) f (x) 1, x, x 2,, x n 1 1, (x a), (x a) 2,, (x a) n 1.?. 26. V F n, σ V 0. σ V., ; V, σ. 27. (1) τ ; (2) σ, τ EndV V, σ + τ V ; «A B (3) A, D, B, C,. C D 28. V F n, A V, P V, σ A, τ P P. σ τ V.,. V, σ τ. 29. V = R 3, σ(x, y, z) = (x + 2y z, y + z, x + y 2z). (1) σ ; (2) σ. 30. V n, U V. W = {α V (α, β) = 0, β U}. W V V = U W (2). 32. V = R[x] n, U = {f(x) V f(0) = 0}. (f(x), g(x)) = Z 1 (1) U V n 1, U ; (2) n = 3, U U. 0 f(x)g(x) d x. 33. R n W, e 1 + e 2 W e : f(x) Fourier a n, b n (n > 0) f(x) cos nx, sin nx 70

43 V V σ, σ 5., V, Imσ. τ, Kerτ = Imσ, Imτ = Kerσ. 37. α 0 V, σ(α) = α 2(α, α 0)α 0, α V. (1) σ ; (2) σ. 38. : V σ ( (σ(α), β) = (α, σ(β))) σ V. 39. σ R 2, «c s P = s c c 2 + s 2 = 1. σ, ( : 1 3.) 41. σ EndR 2. S = {(x, y) : 0 x, y 1} σ G = σ(s) = {σ(x, y) : (x, y) S}. : (1) G ; (2) G, σ? (3) σ, G? 42. (1) σ EndR 2. C (, ). σ(c) ( C ); (2) P n ( C n ), σ(q) ; (3),, σ ; (4) σ EndR 3. Q. σ(q) ( Q ). 43. Householder H v v,. Householder. 44. Givens G. x = (x 1, x 2,, x n ) T, Gx. x, Givens x e (1) ; (2) 2.4.9, 2.4.5; (3). 46. (1) Hermite Hermite? Hermite ; (2), ( 2.4.5) (2.5.3) U V. 49. (2.5.4) (2.5.11) V/U (2.5.12). 71

44 (2.5.15) σ. U σ 57. A M s(f ), B, C M n s(f ). V, σ EndV. σ V «A X 0 B α 1,, α s, α s+1,, α n α 1,, α s, β s+1,, β n «A Y. U = Span{α 0 C 1,, α s } σ, σ σ V/U ᾱ s+1,, ᾱ n β s+1,, β n B C , A, B n. σ F n n, σ : X σ(x) = AXB T. σ E 11,, E 1n, E 21,, E 2n, E n1,, E nn B A. 59. (, ) σ C, ««x x σ : A, y y (x, y) T «a b A =. b a x + yi, σ((x, y) T ) = (a + bi)(x + yi) V = M 2 (C), U = M 2 (R). iu = {A V A = ix, X U}. ( ) U (iu), V/U (V/U) C ,. 64. (2.6.1) C. S? {u k }.? 66. y k+3 2y k+2 5y k+1 + 6y k = 0( k ), σ 29, S = {(x, y, z) T R 3 0 x, y, z 1}. σ(s) AX + XB = C, A A, B = C A, C = A. 72. σ : X AX + XB M n (C). σ A B. 73. U, W V. (1) (U + W )/W U/(U T W ) ; (2) (1). 74. ( ) A F m n. x F m, y F m, f(x, y) = x T Ay. 72

45 f(x, y) F m n, A. F m n B(m, n). (1) B(m, n) F ;( 6 ) (2) B(m, n) ; (3) B(m, n) Hom F (F m F n, F ) = (F m F n ) U W V, α, β V. : (1) α + U = P U (α) + U; (2) (α + U) + (β + W ) = (α + β) + (U + W ); (3) (α + U) (β + W ) α β U + W. 76. tr : X tr(x) M n (F ) F σ(xy ) = σ(y X) σ(i) = n. 73