3 30 5 VI VI. W,..., W r V W,..., W r W + + W r = {v + + v r v W ( r)} V = W + + W r V W,..., W r V W,..., W r V = W W r () V = W W r () W (W + + W + W + + W r ) = {0} () dm V = dm W + + dm W r VI. f n f f λ,..., λ r V = V λ V λr V λ λ f VI. V.9 VI. 4. VI.() 3 C V z C s, t z = s + t (s, t R)
= s z t z s = Re(z), t = Im(z) z = s + t C z z = s t z, w C z = z, z + w = z + w, zw = z w, z = (z 0). z z C z = z z = s + t C z z = zz = s + t z = z A = [a j A = [a j C n a b a =., b =. Cn a n b n a b = t ab = a b + + a n b n C n a b
3 3 a, b, c C n, k C a + b c = a c + b c a (b + c) = a b + a c (ka) b = k(a b) a (kb) = k(a b) a b = b a a a 0. a = 0 a, b, c k R R n C n, n a C n a a = a a + 3. a =, b = 0 () a b () b a () a 3.. V a, b V (a, b) V
4 a, b, c V, k C (a + b, c) = (a, c) + (b, c) (a, b + c) = (a, b) + (a, c) (ka, b) = k(a, b) (a, kb) = k(a, b) (a, b) = (b, a) (a, a) 0 a = 0 3.. V () a V a = (a, a) a () a, b V (a, b) = 0 a b 3. a = 0, b = C 3.5 3.3 a, b V () (a, b) a b () a + b a + b ( ) ( )..5 () a = 0 0 a 0 s, t C
3 5 0 (sa + tb, sa + tb) = a s + (a, b)st + (a, b)ts + b t s = b, t = (a, b) a b 4 (a, b) b + b (a, b) = b ( a b (a, b) ). () a + b = a + (a, b) + (a, b) + b a + (a, b) + b a + a b + b = ( a + b ) V (u,..., u n ) (u, u j ) = δ j V (u,..., u n ) V V C n 3.4 U = (u,..., u n ) V a = a u + + a n u n, b = b u + + b n u n V (a, b) = a b + + a n b n. a b a b U.,. a n b n
. n n (a, b) = (a u + + a nu n, b u + + b nu n) = a b j(u, u j). = j= (u, u j) = δ j = j n (a, b) = a b. = 0 3.5 W V W V (u,..., u r ) W a V W P W (a) = (a, u ) u u + + (a, u r) u r u r P W (a) W a P W (a) W. 7. P W (a) W W b = b u + + b r u r (b, a P W (a)) = (b, a) (b, P W (a)). (b, a) = (b u + + b r u r, a) = b (u, a) + + b r (u r, a). 3.4 ( (b, P W (a)) = b u + + b r u r, (a, u ) u u + + (a, u ) r) u r u r = b (a, u ) + + b r (a, u r ) = b (u, a) + + b r (u r, a). (b, a P W (a)) = 0
3 7 3. 0 a,..., a r V. 7.7 7.5 3.3 3 0 0 0,, 5 4 3 V V f V a, b V (f(a), f(b)) = (a, b) (3.) 3. f V (u,..., u n) V (f(u ),..., f(u n )) V U = (u,..., u n ) V V f U A a, b V U x = [a U, y = [b U a = (u,..., u n )x, b = (u,..., u n )y
8 a, b f f(a) = (f(u ),..., f(u n ))x, f(b) = (f(u ),..., f(u n ))y. A f(a) = (u,..., u n )Ax, f(b) = (u,..., u n )Ay. f (3.) 3.4 Ax Ay = x y t x t AAy = t xy 3. A t AA = E V f (f(a), b) = (a, f (b)) a, b f f V f A f B Ax y = x By t x t Ay = t xb y. x, y B = t A A = t A A
3 9 A A A = E C n 3. n A = [a a n () A () (a,..., a n ) n () x C n Ax = x. () (). t AA = E t a. t a n [a a n = E. (, j) t a a j a a j = δ j () = (). t AA = E Ax = Ax Ax = t x t AAx = t xx = x. () () = (). x, y C n A(x + y) Ax Ay = x + y x y. A(x + y) Ax Ay = (A(x + y)) (A(x + y)) Ax Ax Ay Ay = t (x + y) t AA (x + y) t x t AAx t y t AAy = t x t AAy + t y t AAx. t x t AAy + t y t AAx = x + y x y. t AA (s, t) b st x = e s, y = e t b st + b ts = e s + e t e t e s = δ st, x = e s, y = e t b st + b ts = e s + e t e s e t = 0.
0 t ( t AA) = t AA = t AA b ts = b st. b st + b st = δ st, b st + b st = 0 b st = δ st t AA = E. 3. n A U U AU U AU = D DD = DD AA = (UDU )(UDU ) = UDDU = UDDU = (UDU )(UDU ) = A A 3.3. AA = A A A 3.3 A A = A A = A 3.4 n A A. A n
3 α A V α α b V α A(A b) = A (Ab) = αa b A b V α a V α Aa b = a A b = 0. Aa V α V α A V α (u,..., u m ) V α (u m+,..., u n ) V α (u,..., u n ) V [ αem O (Au,..., Au n ) = (u,..., u n ) O A U = [u u n U [ U αem O AU =. O A [ [ [ ααem O αem O αem O = = (U AU)(U A U) O A A O A O A [ = U AA U = U AA ααem O U = = O A A A A = A A. A n m [ U U A U = D Em O X = U O U [ αem O X AX = O D A α,..., α k A C n A ( 4.) VI., VI. C n = V α V αk
A 9. 3.5 3. ( ) A n α,..., α r A P,..., P r P = P = P ( r), P P j = O ( j), E = P + + P r A = α P + + α r P r A. A C n = V α V αr x C n x = x + + x r (x V α ) =,..., r P x x P P x = P (P x) = P x = x P = P C n y = y + + y k y V α 3.5 P x y = x P y, P x y = x (y + + y k ) = x y = x y. x P y = y = P y. P = P j x, y C n P P j x y = P j x P y = 0.
3 3 P P j = O x C n x = P x + + P rx P + + P r = E. (α P + + α r P r )x = α P x + + α r P r x = A(P x) + + A(P rx) = A(P x + + P rx) = Ax A = α P + + α rp r A = α P + +α r P r W = ImP y = P x W Ay = (α P + + α r P r )P x = α P x = α y y V α. W V α. C n = W W r W = V α. P W P = P. 3.5 0 3.7 A = 0 0 A A = A AA = A = A A A λ F A (λ) = det(λe A) = λ = t 3 + 3t + = (t )(t + ). λ λ = 0 E A = 0. 0 0 0 t t t. λ = t
4 a = λ = E A = 0 0 0. 0 0 0 s t s, t s = s + t 0. t 0 a =, a3 = 0 (a, a3) 0 b =, b 3 = a 3 a = 0 a, b, b 3 u = 3, u =, u 3 = 0 U = [u u u 3 0 0 U AU = 0 0. 0 0 A (e, e, e 3) (e, e, e 3 ) = (u, u, u 3 )P P = [p j U =
3 5 [u u u 3 P = U = U = t u t u t u 3 = 3 3 3 0 V P = [p u p u p 3 u = u t u = 3 3 3 3 V P = [p u + p 3 u 3 p u + p 3 u 3 p 3 u 3 + p 33 u 3 = [u [t u 3 u t u 3 = 3 3 P + P = E A A = 3 3 3 3 3 3 3 3 3 3 3 5. 3 3 5 0 3.4 A = 0 0 0 0 3.8 3. α, β P + P = E, A = αp + βp A αe = αp + βp αp αp = (β α)p. A βe = (α β)p., P = (β α) (A αe), P = (α β) (A βe). 3.5 A α A x A x = αx
3.9 A () A A () A A. A U λ O U AU =.... O λ n λ,..., λ n A D A (U AU) = U A U = U AU D = U AU D = D = D. λ D = D. A = UDU = UD U = (UDU ) = A A A D D D = E. λ = λ λ = λ =. A A U U 3.0 A A A. A P D P AP = D. P = t P, t D = D t A = t (P DP ) = t (P D t P ) = t ( t P ) t D t P = P D t P = P DP = A. A A A A
VII 7 A VII [B VII.5 W,... W r V V = W W r V x = x + + x r (x W ) φ ( r) φ : V V, x x V W () φ V () φ φ = φ, φ = φ V V W W () j (x, y) = 0 x W, y W j φ φ j = 0. 3. () + 3 () 3 () 7 [ t a [ 3. x a, b t x = 0. a b x = 0. b +. 3.3 0,, 3 3 4 3. (f(u ), f(u j )) = (u, u j ) = δ j
8 3. (u,..., u n), (v,..., v n) (u,..., u n) = (v,..., v n)p P P p u (v,..., v n ) 3.4 δ j = (u, u j ) = p p j. (p,..., p n ) 3.3 3.4 0 3 0 0 U = 3 3 3, U AU = 0 3 0. 0 0 0 A = 3 3 3 3 3 3 3 3. 3 3 3 3. 0 0 0 0 0 3 3. 0 3 3 3.5 A A = α P + + α rp r α = α A = α P + + α r P r. x α P x = 0 ( =,..., r). A x = α P x = αx. x V α A(A x) = A (Ax) = αa x. A x V α V α (u,..., u s) (A u, u j ) = (u, Au j ) = α(u, u j ) = αδ j. A u = αu. x V α u,..., u s A x = αx VII.5 () φ (φ (x)) = φ (x ) = x = φ (x). V = W W x = x + (x + + x + x + + + x r) x = x + + x + x + + + x r W. W y = y + y (φ (x), y) = (x, y + y ) = (x, y ) = (x, φ (y)), (φ (x), y) = (x, φ (y)).
VII 9 (x, φ (y)) = (x, φ (y)) x V (x, φ (y) φ (y)) = 0. φ (y) φ (y) V = {0}. φ φ = φ, φ = φ V W = Imφ x = x + x (x W, x W ) φ(x) = φ(x ) + φ(x ). x W = Imφ x = φ(y) φ(x ) = φ(φ(y)) = φ(y) = x. y V φ(y) W (φ(x ), y) = (x, φ(y)) = 0. φ(x ) = 0. φ(x) = x φ W () = x W, y W j (x, y) = (φ (x), φ j (y)) = (x, φ φ j (y)) = (x, 0) = 0.