2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i
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1 [ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A A 1k 0 1 A A 2k A 3k 1 A rk ( 1 ) A rank A x 1,, x n (11) m n A b, x a 11 x a 1n x n = b 1 a m1 x a mn x n = b m a 11 a 1n b 1 A =, b = a m1 a mn 0 b m x 1, x = x n (11) Ax = b A b m (n + 1) Ã = (A b) Ã 1 A 11 0 A A 1k c A A 2k c A 3k c 3 (A c) = 0 1 A rk c r c 1
2 2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã ij A A = n a ik ã ik = k=1 n a kj ã kj a 11 a 12 a 1n a a 0 a 22 a 2n a = 21 a 22 a 2n 22 a 2n = a 11 a 0 a n2 a nn a n1 a n2 a n2 a nn nn f : A B A a B f(a) (1) b B b = f(a) a A f (2) a, a A a a f(a) f(a ) f f : A B a a f(a) f(a ) f(a) = f(a ) a = a f f f : A B, b B f 1 (b) k=1 f 1 (b) = {a A f(a) = b} A f b B f f 1 f 1 : B A f B S f S f 1 (S) = {a A f(a) S} A
3 (2016 3Q N) 3 [ ] K R ( ) C ( ) K n n ( R n C n ) x = (x i ), y = (y i ) K n, λ K K n x + y = (x i + y i ) K n λx = (λx i ) K n (1) ( ) x, y K n x + y = y + x (2) ( ) x, y, z K n (x + y) + z = x + (y + z) (3) ( ) o K n x K n x + o = x (4) ( ) x K n x + x = o x K n (5) ( ) x K n, λ, µ K (λ + µ)x = λx + µx (6) ( ) x, y K n, λ K λ(x + y) = λx + λy (7) ( ) x K n, λ, µ K (λµ)x = λ(µx) (8) ( ) x K n 1 x = x (3) o 0 x = (x i ) (4) x i x i V V x, y V x + y x + y V K V λ K x V λx λx V ( ) V K V K ( ) (1) (8) K R V K C V K n K n K K n V = {o} {o} (1) K m n M(m, n; K) = {(a ij ) a ij K} (2) K n P n (K) = {a 0 +a 1 X+ +a n X n a i K} (3) K K[X] = {a 0 +a 1 X+ +a n X n a i K, n N {0}} (4) K = R R C(R) = {f(x): R R f(x) } (5) K = R R C C (R) = {f(x): R R f(x) C } ( ) V K V r x 1,, x r λ 1 x λ r x r, (λ 1,, λ r K) x 1,, x r
4 4 (2016 3Q N) [ ] V K (R n C n ) V a 1,, a r a 1,, a r W = {λ 1 a 1 + λ 2 a λ r a r λ 1, λ 2,, λ r K} W x λ 1, λ 2,, λ r K x = λ 1 a 1 + λ 2 a λ r a r W ( ) V r a 1, a 2,, a r λ 1 a 1 + λ 2 a λ r a r = o λ 1 = λ 2 = = λ r = 0 a 1, a 2,, a r n λ 1 a 1 + λ 2 a λ r a r = o n r λ 1 A = (a 1 a 2 a r ) x = Ax = o λ r a 1, a 2,, a r Ax = o o ( ) a 1, a 2,, a r K n A = (a 1 a 2 (1) a 1, a 2,, a r (2) Ax = o x = o (3) rank A = r a r ) a 1, a 2,, a r x W x = λ 1 a 1 + λ 2 a λ r a r a 1, a 2,, a r ( ) V r a 1, a 2,, a r a 1, a 2,, a r λ 1 a 1 + λ 2 a λ r a r = o (λ 1, λ 2,, λ r ) (0, 0,, 0) ( ) V r a 1, a 2,, a r a i a 1,, a i 1, a i+1,, a r a 1,, a r V a 1 a 2,, a r a r a 1,, a r 1 a 1 a 2,, a r
5 (2016 3Q N) 5 [ ] K R C V K W V x, y W V x + y V λx V, (λ K) W W V ( ) K V W V W V W V o W x, y W, λ K x + y W, λx W V W o W x, y W, λ K x + y W, λx W V (1) (2) (5) (8) o W 1 K x = ( 1)x W x + ( x) = (1 1)x = o ( ) K V W V (1) x, y W x + y W (2) x W, λ K λx W {o} V V V W V o W o W V W V (1) V = C 0 (R) f, g C 0 (R) f + g λf (f + g)(x) = f(x) + g(x), (λf)(x) = λf(x) f + g, λf R C 0 (R) (x) 0 C C W = C (R) C 0 (R) C C C C (R) C 0 (R) (2) V = Seq(R) {a n } n=1, {b n } n=1 λ {a n } n=1 + {b n } n=1 = {a n + b n } n=1, λ{a n } n=1 = {λa n } n=1 Seq(R) {0} n=1 W = Conv(R) V 0 Conv 0 (R) W V c 1,, c k R a n+k + c 1 a n+k c n a n = 0 {a n } U = {{a n } n=1 Seq(R) a n+k + c 1 a n+k c k a n = 0} V
6 6 (2016 3Q N) [ ] V W 1, W 2 V W 1 + W 2 W 1 + W 2 = {x 1 + x 2 x 1 W 1, x 2 W 2 } W 1 + W 2 V W 1, W 2 W 1 + W 2 x x 1 W 1 x 2 W 2 x = x 1 + x 2 x W 1 + W 2 x 1, x 2 W 1 + W 2 W 1, W 2 W 1 W 2 ( ) V W 1, W 2 W 1 + W 2 = {x 1 + x 2 x 1 W 1, x 2 W 2 } V W 1 + W 2 W 1, W 2 W 1 + W 2 x 1 + x 2 = x 1 + x 2 (x i, x i W i) x i = x i W 1 + W 2 W 1 W 2 W 1 W 2 V W 1, W 2 W 1 W 2 = {x V x W 1 x W 2 } V W 1, W 2 W 1 W 2 V V K V a 1,, a k V W = {λ 1 a λ k a k λ i K} V a 1,, a k a 1,, a k a 1,, a k = {λ 1 a λ k a k λ i K} V W 1, W 2 W 1 = a 1,, a n, W 2 = b 1,, b m x W 1 a 1,, a n y W 2 b 1,, b m W 1 + W 2 a 1,, a n, b 1,, b m W 1 + W 2 = a 1,, a n, b 1,, b m a 1,, a n b 1,, b m W 2 a 1,, a n, b 1,, b m x W 1 W 2 x W 1 x W 2 λ 1,, λ n K µ 1,, µ m K x = λ 1 a λ n a n = µb µ m b m λ 1,, λ n µ 1, µ m λ 1 a λ n a n (µb µ m b m ) = o λ 1,, λ n ( µ 1,, µ n ) W 1 W 2
7 (2016 3Q N) 7 [ ] W V W a 1,, a r V W = a 1,, a r W x x = λ 1 a λ r a r (λ 1,, λ r K) a 1,, a r x K R C V K ( ) ( ) V a 1,, a n V (1) a 1,, a n (2) V a 1,, a n V R n e 1,, e n λ 1 λ 1 e λ n e n = λ λ n 0 = = o 0 1 λ 1 = = λ n = 0 e 1,, e n x = (x i ) R n x = x i e i R n e 1,, e n e 1,, e n R n R n R n f 1,, f n f i = e e i (1 i n) f 1,, f n R n V a 1,, a n ( ) a 1,, a n V x V λ n x = λ 1 a λ n a n x V x = λ 1 a λ n a n λ 1,, λ n x = µ 1 a µ n a n i λ i = µ i V V ( ) K V {o} n V n dim K V = n V = {o} dim K V = 0 K dim V W V W W dim W a 1 o W = a 1 R n W = {ta 1 t R} a 1 a 1 W dim W = 1
8 8 (2016 3Q N) V W W = a 1,, a r a 1,, a r W ( ) V W r dim W r W (dim W + 1) W V W W ( ) W {o} V a 1,, a s W a s+1,, a r W a 1,, a r W W V W V W V dim W dim V W r = dim W a 1,, a r a 1,, a r W x W a 1,, a r a 1,, a r = W ( ) W V (1) dim W dim V (2) dim W = dim V W = V ( ) V n (1) n + 1 (2) n W 1, W 2 V dim W 1 = r, dim W 2 = s W 1, W 2 W 1 = a 1,, a r, W 2 = b 1,, b s W 1 + W 2 W 1 + W 2 = a 1,, a r, b 1,, b s dim(w 1 + W 2 ) r + s = dim W 1 + dim W 2 ( ) W 1, W 2 V dim(w 1 + W 2 ) = dim W 1 + dim W 2 dim(w 1 W 2 )
9 (2016 3Q N) 9 [ ] K R C V K V x, y V λ K x + y V λx K V, V K f : V V V V x V f(x) V x, y V, λ K x + y V f(x + y) V λx V f(λx) V f(x), f(y) V f(x) + f(y) V λf(x) V (1) f(x + y) = f(x) + f(y), (2) f(λx) = λf(x) f V = V ( ) K V V f : V V x, y V, λ K (1) f(x + y) = f(x) + f(y), (2) f(λx) = λf(x) V = V (1) V = R n, V = R m A m n f : R n R m f(x) = Ax x, y V λ R f(x + y) = A(x + y) = Ax + Ay = f(x) + f(y) f(λx) = A(λx) = λax = λf(x) f (2) V = C (R) d dx : V V f C (R) d dx (f + g) = (f + g) = f + g = d dx f + d dx g d dx (λf) = (λf) = λf = λ d dx f d dx f = df dx d dx f : V V V a 1,, a n x V x = λ 1 a λ n a n f f(x) = λ 1 f(a 1 ) + + λ n f(a n ) V a 1,, a n f f(a 1 ),, f(a n ) x V f
10 10 (2016 3Q N) V K id V : V V id V (x) = x id V V ( ) K V, V f : V V g : V V g f = id V, f g = id V V V V = V V V f : V V (g : V V ) V V f : V V g : V V g f = id V, f g = id V x V x = g(x ) f g = id V f(x) = f(g(x )) = id V (x ) = x f f(x) = o g f = id V x = g(f(x)) = g(o) = o f f : V V f : V V f g : V V x, y V x = g(x ), y = g(y ) f f(x + y) = f(x) + f(y) = f(g(x )) + f(g(y )) = x + y g(x ) + g(y ) = x + y = g(f(x + y)) = g(x + y ) λ K f(λx) = λx λx = g(f(λx)) = g(λx ) λg(x ) = g(λx ) g ( ) V, V K f : V V f V V K = R A m n B l m f : R n R m, g : R m R l f(x) = Ax, g(y) = By g f g f(x) = g(f(x)) = B(f(x)) = B(Ax) = (BA)x l = m = n g f = id R n x R n R n e 1,, e n x = g f(x) = (BA)x (e 1 e n ) = (BA)(e 1 e n ) BA = I n B A f(x) = Ax A
11 (2016 3Q N) 11 [ ] V K n V e 1,, e n e 1,, e n V x V x = a 1 e 1 + a 2 e a n e n (a i K) x V a K n a 1 V x = a 1 e 1 + a 2 e a n e n a = K n a n E = (e 1,, e n ) V K n φ E φ E (a 1 e 1 + a 2 e a n e n ) = x = a 1 e a n e n, y = b 1 e n + + b n e n a 1 + b 1 φ E (x + y) = φ E ((a 1 + b 1 )e (a n + b n )e n ) = a n + b n a 1 b 1 = + = φ E (x) + φ E (y) a 1 a n a n b n c K ca 1 φ E (cx) = φ E (ca 1 e ca n e n ) = ca n a 1 = c = cφ E (x) a n φ E φ V K n x = a 1 e 1 + a 2 e a n e n V K n a 1 a = a n x E V K dim V
12 12 (2016 3Q N) F = (f 1,, f n ) V x = a 1 e a n e n F x = b 1 f 1 + b 2 f b n f n F φ F : V K n b 1 b = φ F (x) = F b f j V E = (e 1,, e n ) b n f j = p 1j e p nj e n e i ( ) p 1j f j = (e 1 e n ) p nj p 11 p 12 p 1n p (f 1 f n ) = (e 1 e n ) 21 p 22 p 2n p n1 p n2 p nn F x = b 1 f b n f n E p b 11 p 12 p 1n 1 b p x = (f 1 f n ) = (e 1 e n ) 21 p 22 p 2n 1 b n b p n1 p n2 p n nn P = (p ij ) E F a 1 x V E a n a 1 x = (e 1 e n ) E p a 11 p 12 p 1n 1 b p = 21 p 22 p 2n 1 a n b p n1 p n2 p n nn a n
13 (2016 3Q N) 13 [ ] V, W K V E = (e 1,, e n ) W F = (f 1,, f m ) E V x V x 1 x = x 1 e x n e n = (e 1 e n ) F W y W y = y 1 f y m f m = (f 1 f m ) f : V W y = f(x) e j V f(e j ) W a 1j f(e j ) = a 1j f a mj f m = (f 1 f m ) a mj f x V f(x) = f(x 1 e x n e n ) = x 1 f(e 1 ) + + x n f(e n ) x 1 a 11 a 1n x 1 = (f(e 1 ) f(e n )) = (f 1 f m ) x n a m1 a mn x n y = f(x) y 1 a 11 a 1n x 1 = y m a m1 a mn x n A = (a ij ) V E W F f E V φ E V K n F W φ F W K m f K n K m f : K n K m f A f A x n y 1 y m V φ E K n f W φ F f A K m x = x 1 e x n e n φ E x = x 1 f f A A f(x) φ F x 1 = A x x n x n
14 14 (2016 3Q N) V E = (e 1,, e n) W F = (f 1,, f m) x V, y W E, F x 1 x = x 1e x ne n = (e 1 e n) y = y 1f y mf m = (f 1 f m) f E F x V E x, y = f(x) F ỹ ỹ = A x A E E P F F Q E = EP, F = F Q x = E x V E x = EP x f(x) = y = F ỹ x n y 1 y m f F AP x = F Q 1 AP x ỹ = Q 1 AP x f E, F Q 1 AP f V f : V V = V V V (F = E) a 11 a 1n f(x 1 e x n e n ) = (e 1 e n ) a n1 a nn A = (a ij ) f E E E P f E P 1 AP n A, A P A = P 1 AP A A f A f A x 1 x n V f W x K n φ E φ E φ F φ F P K n x f A K m f y = f(x) Q 1 K m φ E φ E φ F φ F P x = P x f A ỹ = AP x Q 1 ỹ = Q 1 AP x
15 (2016 3Q N) 15 [ ] V, W K f : V W f (1) x, x V f(x + x ) = f(x) + f(x ) (2) x V, λ K f(λx) = λf(x) V x f o f Ker f W y f f Im f Ker f V f (1) f(o) = o o Ker f (2) x, x Ker f x + x Ker f (3) x Ker f, λ K λx Ker f f(x + x ) = f(x) + f(x ) = o + o = o f(λx) = λf(x) = λo = o Ker f V Im f W f (1) o = f(o) Im f (2) y, y Im f Im f y = f(x), y = f(x ) x, x V y + y = f(x) + f(x ) = f(x + x ) x + x V y + y Im f (3) y Im f, λ K y = f(x) x V λy = λf(x) = f(λx) λx V λy Im f Im f W ( ) f : V W f Ker f V Ker f = {x V f(x) = o} V f Im f W Im f = {y W y = f(x) x V } W = {f(x) W x V } W
16 16 (2016 3Q N) V, W V W f : V W V x (V ) x 1 y 1 x = f( x) (W ) ỹ = m n x n A ỹ = f A ( x) = A x Ker f A A x = o A x = o r = rank A A x = o n r x = t 1 ũ t n r ũ n r, (t 1,, t n r K, ũ i K n ) ũ i V u i V Ker f = u 1,, u n r {u 1,, u n r } Ker f dim(ker f) = n r Im f A A x A = (ã 1 ã n ) Im f A = {x 1 ã x n ã n x i K} = ã 1,, ã n ã 1,, ã n Im f A A = (ã 1 ã n ) r = rank A dim(im f A ) = r ã i W a i W Im f W ( ) f : V W A dim(im f) = rank A ( ) V, W f : V W dim(ker f) + dim(im f) = dim V f : V W f y W f(x) = y x V f Im f = W f x Ker f f(x) = f(o) = o f Ker f = {o} y m
17 (2016 3Q N) 17 [ ] K R C V K V a, b a, b ( ) V a b (a, b) K (a, b) a b (1) ( ) (a, b) = (b, a), (2) ( ) (a 1 + a 2, b) = (a 1, b) + (a 2, b), (3) ( ) (λa, b) = λ(a, b) = (a, λb), (λ K), (4) ( ) (a, a) 0 (a, a) = 0 a = o K = R (1), (3) (1) ( ) (a, b) = (b, a), (3) ( ) (λa, b) = λ(a, b) = (a, λb), (λ R), K = R K = C V = K n a = (a i ), b = (b i ) (a, b) = a 1 b 1 + a 2 b a n b n = n a i b i = t ba = b a i=1 (a, b) K n K n K = C V a V (a, a) a V ( ) ( ) a = (a i ) V a (a, a) V = K n a = (a, a) = a a a n 2 ( ) a, b V λ K (1) ( ) a 0 a = 0 a = o (2) ( ) λa = λ a (3) ( ) (a, b) a b (4) ( ) a + b a + b
18 18 (2016 3Q N) x, y V (x, y) = 0 x y x y r x 1,, x r V x 1,, x r x 1,, x r x 1,, x r 1 ( ) x, y (x, y) = 0 x 1,, x r x i o x i x j (x i, x j ) = 0 x 1,, x r x i = 1 x 1,, x r x 1,, x r V x 1,, x r V 1 a, b V (a, b) a b 1 ( ) V a, b V θ (0 θ π) θ = cos 1 (a, b) a b V W V W W = {a V x W (a, x) = 0} V W W V = W + W x W W x W x W 0 (x, x) = 0 x = o V = W + W W W = {o} V W W V = W W x V dim W + dim W = dim V x = a + b, ( a W, b W ) W, W 1, W 2 V (1) (W ) = W (2) (W 1 + W 2 ) = W 1 W 2 (3) (W 1 W 2 ) = W 1 + W 2 V = K n a 1,, a r K n W = a 1,, a r W a i x W i (a i, x) = t a i x = 0 A = (a 1 a r ) W Ax = o
19 (2016 3Q N) 19 [ ] V V (x, y) a, b V 1 a, b θ b a ( ) h h = b cos θ cos θ = h = b cos θ a a (b, a) a b h = (b, a) a 2 a b = b h = b (b, a) a 2 a (b, a) = ( b ) (b, a) a 2 a, a = (b, a) (b, a) 2 (a, a) = 0 a b a V b a 1 {a 1,, a r } a 1,, a r (1) v 1 = a 1 (2) v 1,, v i 1 v i v i = a i i 1 k=1 (a i, v k ) v k 2 v k = a i (a i, v 1 ) ( ) (3) i u i = v i ( ) v i v 1 2 v 1 (a i, v i 1 ) v i 1 2 u 1,, u r a 1,, a r W u 1,, u r W ( ) V {o} v i 1
20 20 (2016 3Q N) [ ] n A A A = AA = I A A = A ( ) A A A = AA = I A A = A A t AA = A t A = I A t A = A V K a, b V n A (a, Ab) = (Ab) a = t (Ab)a = t b t Aa = b A a = (A a, b) (A ) = A (Aa, Ab) = (a, A Ab) A (Aa, Ab) = (a, b) A (Aa, b) = (a, Ab) ( ) A A A A A A = ±1 T n T f = f T : R n R n a, b R n (T a, T b) = (a, b) (f(a), f(b)) = (a, b) f R n R n U f = f U : C n C n ; f U (x) = Ux C n C n ( ) V f a, b V (f(a), f(b)) = (a, b) f K = R T R n C n K = R (U ) ( ) U (1) U (2) a, b V (Ua, Ub) = (a, b) (3) U = (u 1 u n ) u i = 1 i j (u i, u j ) = 0 (4) U = (u 1 u n ) u 1,, u n
21 (2016 3Q N) 21 [ ] V K T : V V λ K T (x) = λx o x V λ T x λ T λ W λ = {x V T (x) = λx} V W λ λ W λ λ V W T (W ) W W T V {o} T V x x T T λ : V V T λ (x) = T (x) λx Ker(T λ ) = {x V T λ (x) = o} = {x V T (x) λx = o} = W λ Ker(T λ ) λ i 1 W (i) λ = x V i {}}{ T λ T λ (x) = o T λ T (x) = T λ (T (x)) = T (T (x)) λt (x) = T (T (x) λx) = T T λ (x) W (i) λ V T V V = W 1 W 2 W k x = w 1 w 2 w k (w i W i ) T (x) = T (w 1 w 2 w k ) = T (w 1 ) + T (w 2 ) + + T (w k ) T (W i ) W i T (w i ) W i T (x) = T (w 1 ) T (w 2 ) T (w k ) W 1 W 2 W k T
22 22 (2016 3Q N) n A K n T A (x) = Ax A T A Ax = λx o x K n λ A x λ λ W λ W λ = {x K n Ax = λx} = {x K n (A λi n )x = o} λ W λ {o} dim W λ 1 A n A λ o x Ax = λx (A λi n )x = o x o A λi n A λi n = 0 A λi n = 0 (A λi n )x = o o A λi n = 0 A ( ) A n λ A λ A λi n = 0 x λ (A λi n )x = o o xi n A x A xi n A = 0 A A K = C A φ A (x) = xi n A φ A (x) = 0 ( A ) λ 1,, λ k φ A (x) φ A (x) = (x λ 1 ) m 1 (x λ k ) m k m i λ i W (i) λ x W (i) λ W (i) λ = {x Kn (A λi n ) i x = o} (A λi n ) i+1 x = (A λi n ) ( (A λi n ) i x ) = (A λi n )o = o x W (i+1) λ {o} W λ = W (1) λ W (i) λ W (i+1) λ K n n m W (m) λ = W (m+1) λ = W (m+2) λ = W (m) λ λ λ m W λ W λ = W (m) λ
23 (2016 3Q N) 23 [ ] A n A λ 1,, λ k λ i W λi V V = W λ1 W λk W λi m i W λi a i1,, a imi A Aa ij W λi W λi V a 11,, a 1m1, a 21,, a 2m2,, a kmk A B 1 W λi λ i B i = V P λ 1 m 1 P 1 λ AP = 1 λ 2 B k λ i λk A φ A (x) = xi A φ A (x) x n P φ A (x) = x n + a 1 x n a n 1 x + a n φ P 1 AP (x) = xi P 1 AP = P 1 (xi A)P = P 1 xi A P = xi A = φ A (x) φ P 1 AP (x) = φ A (x) = x n + a 1 x n a n 1 x + a n P 1 AP x = A φ P 1 AP (x) = (x λ 1 ) m 1 (x λ k ) m k φ P 1 AP (A) = (A λ 1 I) m 1 (A λ k I) m k = A n + a 1 A n a n 1 A + a n I = 0 φ P 1 AP (x) = φ A (x) φ A (A) = A n + a 1 A n a n 1 A + a n I = 0
24 24 (2016 3Q N) [ ] A n K n A K n a 1,, a n λ i Aa i = λ i a i a 1,, a n n P = (a 1 a n ) a 1,, a n P a i AP = A(a 1 a n ) = (Aa 1 Aa n ) = (λ 1 a 1 λ n a n ) λ 1 λ 1 = (a 1 a n ) = P P 1 AP = λ 1 λ n {a 1,, a n } P = (a 1 a n ) P 1 AP A {a 1,, a n } A P P 1 AP A A A P A ( ) n A (1) A n (2) A λ λ W λ A n a 1,, a n P = (a 1 a n ) P 1 AP A λ m λ (A λi)x = o W λ 1 W λ m + 1 W λ V λ m + 1 λ x λ m + 1 λ W λ m A λ 1,, λ k m 1,, m k dim W λi = m i W λi m i ( ) a i1,, a imi a 11,, a 1m1, a 21,, a kmk n P = (a 11 a 1m1 a 21 a kmk ) P 1 AP P A xi A = 0 1 dim W λ m = 1 n A n A A λ n λ n
25 [ ] n A (2016 3Q N) 25 (1) (2) (3) (4) (5) (1) x n A xi n = 0 A xi n A xi n = ( 1) n (x λ 1 ) m 1 (x λ k ) m k A xi n = 0 λ 1,, λ k A xi n x n x n ( 1) n A xi n n x xi n A ( 1) n (2) λ i W λi (A λ i I n )x = o λ i A λ i I n 1 A 11 0 A A 1k A A 2k A A λ i I 3k A ri k (A λ i I n )x = o x = c 1 x i1 + + c ri x iri x i1,, x iri W λi (1) (A λ i I n )x = o 1 m i (2) x ij Ax ij = λ i x ij A λ i I n A λ i I n n m i
26 26 (2016 3Q N) (3) λ i m i dim W λi (1) (x λ 1 ) m 1 (x λ k ) m k λ i m i W λi (2) r i n rank(a λ i I n ) i r i = dim W λi = m i A i r i m i A (4) P 1 AP n P (2) x ij P P = (x 11 x 1m1 x 21 x 2m2 x k1 x kmk ) A ( ) W λi W λi x i1,, x imi u i1,, u im1 P = (u 11 u 1m1 u 21 u 2m2 u k1 u kmk ) ( ) P (P P = I n ) (AA = A A ) (5) (4) P P 1 AP P P P 1 AP = λ 1 λ1 λ 2 λ k λ 1 AP = P λ k
27 (2016 3Q N) 27 [ ] K R C A n A K n A ( ) ( ) u 1 A λ 1 u 1 = 1 u 1 K n {u 1, p 2,, p n } U 1 = (u 1 p 2 p n ) ( ) U1 1 AU 1 = U1 1 (λ λ1 1u 1 Ap 1 Ap n ) = o A 1 A 1 n 1 A 1 u 2 λ 2, u 2 = 1 u 2 K n 1 U 2 ( ) (U 2) 1 AU 2 λ2 = o A 2 ( ) 1 o U 2 = o U 2 ( ) ( ) ( ) U2 1 o 1 o 1 o U 2 = o (U 2) o U 2 = o (U 2) = I U n 2 U 2 ( ) ( ) ( ) U2 1 U o AU λ1 1 o 1U 2 = o (U 2) 1 o A 1 o U 2 = U = U 1 U 2 U n 1 λ 1 λ U 1 AU = 2 λ n λ 1 λ 2 A 2 U i U A ( ) A n U U 1 AU U 1 AU U U 1 A U = U A U = (U AU) = (U 1 AU) = λ 1 λ n U 1 AA U = (U 1 AU)(U 1 A U) = (U 1 A U)(U 1 AU) = U 1 A AU AA = A A
28 28 (2016 3Q N) ( ) A A A = AA ( ) A A A A A = AA P P = P P = I P P 1 AP = P AP A α i W αi x i1 x imi (m i α i ) P = (x 11 x 1m1 x 21 x 2m2 x k1 x kmk ) P 1 AP W αi x ij u i1,, u imi U = (u 11 u 1m1 u 21 u 2m2 u k1 u kmk ) U U 1 AU = U AU A i j W αi W αj i j a, b (u ia, u jb ) = 0 (1) H H = H A A = t A = A (2) U U U = UU = I A t AA = A t A = I ( ) ( ) a b a (3) A = (a, b ) AA b a = A A = 2 + b a 2 + b 2 ( ) 1 1 A = 1 1 ( ) ( ) ( ) ( ) ( ) ( )
29 (2016 3Q N) 29 [ ] A A A n α = a + bi A α W α v 1,, v s A Av = Av = αv α α A v α (v i, v j ) = t (v j )v i = t v j v i = (v i, v j ) = 0 v j 2 = (v j, v j ) = (v j, v j ) = 1 v 1,, v s α W α α α α j (v j, v j ) = 0 u 2j 1 = 1 2 (v j + v j ), u 2j = i 2 (v j v j ) u 2j 1, u 2j (u 2j 1, u 2j ) = u 2j 1 = u 2j = ( 1 2 (v j + v j ), ( 1 2 (v j + v j ), ) i (v j v j ) 2 ) 1 (v j + v j ) 2 = i 2 (v j + v j, v j v j ) = 0, = 1 2 (v j + v j, v j + v j ) = 1, ( ) i i (v j v j ), (v j v j ) = (v j v j, v j v j ) = 1 Au 2j 1 = 1 2 A(v j + v j ) = 1 2 (αv j + αv j ) = au 2j 1 + bu 2j Au 2j = i 2 A(v j v j ) = i 2 (αv j αv j ) = bu 2j 1 + au 2j ( ) a b A(u 2j 1 u 2j ) = (au 2j 1 + bu 2j bu 2j 1 + au 2j ) = (u 2j 1 u 2j ) b a
30 30 (2016 3Q N) A λ 1,, λ k, α 1 = a 1 + b 1 i,, α r = a r + b r i α 1,, α r {v 1,, v r } u 2j 1 = 1 (v j + v j ), u 2j = i (v j v j ) 2 2 {u 1, u 2,, u 2r 1, u 2r } λ 1,, λ k p 1,, p k P P = (p 1 p k u 1 u 2 u 2r 1 u 2r ) P P ( ) A P P 1 AP λ i, a j, b j P 1 AP = λ 1 λ k a 1 b 1 b 1 a 1 a r b r b r a r ( ) α = a + bi 2 a b b a (1) A k = n, r = 0 (2) A λ 1 = = λ k = 0, a 1 = = a r = 0, b j (3) A θ j λ i = ±1, a j = cos θ j, b j = sin θ j
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(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/98/136512264.jpg "(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/101/151465237.jpg "(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
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LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y
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