(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 c p = ca cb p c a p = c q = b d p q = ac + bd p p = a 2 + b 2 0 p p p p p q θ p q = p q cos θ 1

2 (2016 2Q H) a, b (a, b) a a f : R 2 R 2 x y = f(x) g : R 2 R 2 f + g : R 2 R 2 (f + g)(x) = f(x) + g(x) f g k R kf : R 2 R 2 (kf)(x) = kf(x) f k R 2 ( ) X = (x 1, x 2 ) Y = (y 1, y 2 ) F : R 2 R 2 { y1 = ax 1 + bx 2 + e 1 y 2 = cx 1 + dx 2 + e 2 a, b, c, d, e 1, e 2 (e 1, e 2 ) X x1 x = (e x 1, e 2 ) Y ( 2 ) ax1 + bx 2 cx 1 + dx 2 f : R 2 R 2 x1 y1 x = y = = f(x) x 2 y 2 { y1 = ax 1 + bx 2 y 2 = cx 1 + dx 2 a b x 1, x 2 f A = c d s t g : R 2 R 2 x1 B = x = u v x 2 ax1 + bx (f + g)(x) = f(x) + g(x) = 2 sx1 + tx + 2 cx 1 + dx 2 ux 1 + vx 2 (a + s)x1 + (b + t)x = 2 (c + u)x 1 + (d + v)x 2 f + g A B a + s b + t f A g B c + u d + v a + s b + t A + B = c + u d + v

(2016 2Q H) 3 k kf ax1 + bx (kf)(x) = kf(x) = k 2 kax1 + kbx = 2 cx 1 + dx 2 kcx 1 + kdx 2 ka kb kf f A A k kc kd ka kb ka = kc kd f : R 2 R 2, g : R 2 R 2 1 f, g a b s t A =, B = c d u v f g g f x1 x = x 2 (g f)(x) = g(f(x)) ax1 + bx f(x) = 2 cx 1 + dx 2 s(ax1 + bx (g f)(x) = g(f(x)) = 2 ) + t(cx 1 + dx 2 ) u(ax 1 + bx 2 ) + v(cx 1 + dx 2 ) (sa + tc)x1 + (sb + td)x = 2 (ua + vc)x 1 + (ub + vd)x 2 g f sa + tc sb + td ua + vc ub + vd g B f A sa + tc sb + td B A = ua + vc ub + vd B A BA s t a b sa + tc sb + td BA = = u v c d ua + vc ub + vd B A BA

4 (2016 2Q H) a b A = c d A (a b) ( A ) 1 (c d) A 2 a b A A 1 A 2 A c d i j ( A ) (i, ( j) ) a b s t 2 A =, B = A B c d u v a + s b + t A + B = c + u d + v k A k ka kb ka = = Ak kc kd B A sa + tc sb + td BA = ua + vc ub + vd BA (i, j) B i ( ) A j 1 A f x1 x = x 2 ax1 + bx f(x) = 2 a b x1 = cx 1 + dx 2 c d x 2 f(x) = Ax 0 0 O 0 0 A, B, C 1 0 I 0 1 (AB)C = A(BC) (A + B)C = AC + BC, A(B + C) = AB + AC AB = BA A O B O AB = O

[ ] R n n x = (2016 2Q H) 5 R m y = f(x) y 1 = a 11 x 1 + a 12 x 2 + + a 1n x n y 2 = a 21 x 1 + a 22 x 2 + + a 2n x n y m = a m1 x 1 + a m2 x 2 + + a mn x n x 1,, x n x 1 x n a 11 a 12 a 1n a A = 21 a 22 a 2n a m1 a m2 a mn A A f y = f(x) = a 11 x 1 + a 12 x 2 + + a 1n x n a 21 x 1 + a 22 x 2 + + a 2n x n a m1 x 1 + a m2 x 2 + + a mn x n R n f A f(x) = A(x) = Ax a 11 a 12 a 1n a y = Ax = 21 a 22 a 2n a 11 a 12 a 1n A x A x f : R n R m g : R n R m A, B (f + g)(x) = f(x) + g(y) A + B A B A B m A B n h: R m R l C (h f)(x) = h(f(x)) CA C A m h f : R n R l CA l n x 1 x n

6 (2016 2Q H) mn () {a ij 1 i m, 1 j n} m n a 11 a 12 a 1n a A = 21 a 22 a 2n a m1 a m2 a mn m n m n (m, n) m n i A i j A j A i j a ij A (i, j) (i, j) a ij (a ij ) m = n n 2 A = (a ij ), B = (b ij ) i, j a ij = b ij A B A = B A, B A, B m n A = (a ij ), B = (b ij ) A + B = (a ij + b ij ) m n A = (a ij ) c A c ca = (ca ij ) c A, B A B AB A = (a ik ) l m B = (b kj ) m n AB (i, j) c ij c ij = a i1 b 1j + a i2 b 2j + + a im b mj a 11 a 1m i a i1 a ik a im a l1 a lm j a 11 a 1j a 1n a kj = (c ij ) a m1 a mj a mn 0 O 0 o (i, i) 1 0 1 0 0 0 0 0 0 1 I

(2016 2Q H) 7 [ ] A = (a ij ) m n a 11 a 1j a 1n A = a i1 a ij a in a m1 a mj a mn A (i, j) (j, i) n m A t A A A a 11 a i1 a m1 a 11 a 1j a 1n t A = a 1j a ij a mj, A = a i1 a ij a in a 1n a in a mn a m1 a mj a mn ( a + bi = a bi, i = 1) t A A A A = t A ( ) A, B (1) t (AB) = t B t A (2) (AB) = B A ( ) A (1) A t A = A (2) A t A = A (3) A A = A (4) A A = A A = (a ij ) i = j a ii A 0 ( ) A A 0 a 1 O A = O a n A = (a ij ) i > j a ij = 0 A i < j a ij = 0 A a 11 a 1n O a nn, a 11 O a n1 a nn

8 (2016 2Q H) [ ] ( ) n A AX = I XA = I n X X A A 1 () a 0 ax = b x b a n A AX = B X X = IX = (A 1 A)X = A 1 B Y A = B Y Y = Y I = Y (AA 1 ) = BA 1 1 1 0 1 X Y A =, B = A 1 = 0 1 1 0 1 1 AX = B X X = A 1 1 1 B = Y A = B 0 1 1 0 Y Y = BA 1 0 1 = 1 1 A AX = B XA = B X X n A R n R n f(x) = Ax n I n R n R n id R n A AX = XA = I n X X g f g = g f = id R n R n R n g f g f ( ) A, B (1) A 1 (A 1 ) 1 = A (2) AB (AB) 1 = B 1 A 1 (3) A (A ) 1 = (A 1 )

(2016 2Q H) 9 [] m n A A = b 1 b 2 {}}{{}}{ b k {}}{ a 1 { A 11 A 12 A 1k a 2 { A 21 A 22 A 2k a l { A l1 A l2 A lk i A i1,, A ik a i j A 1j,, A lj b j a i = m, b j = n m n B A A B A 11 + B 11 A 12 + B 12 A 1k + B 1k A A + B = 21 + B 21 A 22 + B 22 A 2k + B 2k A l1 + B l1 A l2 + B l2 A lk + B lk n s B A B = b 1 { B 11 B 12 B 1p b 2 { B 21 B 22 B 2p b k { B k1 B k2 B kp j A j A B AB (i, j) A i1 B 1j + A i2 B 2j + + A ik B kj n i a i = b i a 1 a 2 a k {}}{{}}{{}}{ a 1 { A 11 A 12 A 1k a A = 2 { A 21 A 22 A 2k a k { A k1 A k2 A kk

10 (2016 2Q H) A = (a ij ) m n A i A a (i) = ( a i1 a i2 a in ) A = a (1) a (2) a (m) A n l B = (b jk ) k b 1k b b k = 2k b nk B B = ( b 1 b 2 b l ) B a (1) (1) ( x 1 x m ) A = ( x 1 x m ) = x 1 a (1) + + x m a (m) a (m) x 1 x 1 (2) B = ( b 1 b l ) = x 1 b 1 + + x l b l x l (3) AB = A ( b 1 b l ) = ( Ab 1 Ab l ) a (1) (4) AB = B = a (m) a (1) B a (m) B x l a (1) a (1) b 1 a (1) b l (5) AB = ( b 1 b l ) = a (m) a (m) b 1 a (m) b l j 1 0 e j 0 1 1 0 e 1 =, e 2 = 0, 0 0 n I n I n = ( e 1 e 2 e n )

(2016 2Q H) 11 [ ] I n n I ij (i, j) 1 0 n ( ) P ij = I n I ii I jj + I ij + I ji = 1 O 0 1 1 0 O 1 1 O Q i (c) = I n + (c 1)I ii = c (c 0), O 1 1 O 1 c R ij (c) = I n + ci ij = (i j) 1 O 1, 1 P ij P ij = I n, R ij (c)r ij ( c) = R ij ( c)r ij (c) = I n c 0 Q i (c)q i (1/c) = Q i (1/c)Q i (c) = I n A P B = P A P 1 A = P 1 B A P A = B

12 (2016 2Q H) a 1 A n m A = ( ai ) a n P ij A = i j a j a i, Q i (c)a = i ca i, R ij (c)a = (1) i j (r i r j ) (2) i c 0 (cr i ) (3) i j c (r i + cr j ) i j a i + ca j a i a j r i r j a j a i, a i cr i ca i, a i a j r i +cr j a j a i + ca j a j A (1) i j (c i c j ) (2) i d 0 (dc i ) (3) j i c (c j + dc i ) A, B AX = B P (P ij, Q i (c), R ij (c) ) P AX = P B A, B A, B (A B) P (A B) = (P A P B) (A B) P ( ) P P 1 (A B) AX = B A A AX = B A X = B A X = B

[ ] A (1) i j (2) i c 0 (3) i j c (2016 2Q H) 13 ( ) A = (A ij ) A ij m n A mi + 1 (m + 1)i c 0 A i c k = 1,, m A mi + k mj + k A i j A i1 A it ca i1 ca it, A i1 A j1 A it A jt A j1 A i1 A jt A it P = i j O i I m C j I m O A A i1 A it A j1 A jt A i1 + CA j1 A it + CA jt A j1 A jt P A A 1i A si A 1j A sj A 1i A si A 1j + A 1i C A sj + A si C C

14 (2016 2Q H) [ ] A m n A A (141) t o 1 A 11 0 A 12 0 0 A 1r 0 t o 1 A 22 0 0 A 2r 0 t o 1 0 A 3r 0 1 A rr O O O A ij t o A ij 0 A = (a ij ) m n A o j A A = (O a j a n ) a j o a ij 0 i 1/a ij (3) a 1j = = ( a i 1 j = a i+1 j = = a mj = 0 t ) o 1 i 1 A O o A 1 A 1 t o 1 A 11 A 12 A 1r 0 t o 1 A 22 A 2r 0 t o 1 A 3r 0 1 A rr O O O 1 1 o (141) ( ) P 1,, P k P k P 2 P 1 A P = P k P 2 P 1 A P P A A B A B B A A (141) A r A rank A

[ ] x 1, x x (151) (2016 2Q H) 15 a 11 x 1 + a 12 x 2 + + a 1n x n = c 1 a 21 x 1 + a 22 x 2 + + a 2n x n = c 2 a m1 x 1 + a m2 x 2 + + a mn x n = c m m n A = (a ij ), n x = (x i ) m c = (c i ) Ax = c A (151) A c (A c) (151) Ax = c x A A 1 A 1 x = (A 1 A)x = A 1 c A 1 (A c) A 1 A (A c) A Ax = c x (A c) (152) (A c) (A d) = 1 A 11 0 A 12 0 0 A 1r d 1 0 O 1 A 22 0 0 A 2r d 2 0 O 1 0 A 3r d 3 0 1 A rr d r O O O d () P Ax = c P Ax = P c A x = d A x = d x Ax = c x A x = d A x = d x Ax = c d o Ax = c d = o Ax = c A x = d

16 (2016 2Q H) d = o (A d) i 1 = 1 < i 2 < < i r x i1 x 1 x = x i2 x ir x r A x = d x i1 = d 1 A 11 x 1 A 12 x 2 A 1r x r x i2 = d 2 A 22 x 2 A 2r x r x ir = d r A rr x r A = (a ij ) x = x i1 x 1 x i2 x 2 x ir x r d 1 o d 2 = ọ d r o d 1 A 11 x 1 A 12 x 2 A 1r x r x 1 d 2 A 22 x 2 A 2r x r = x 2 d r A rr x r + j i 1,,i r x j a 1j o a 2j ọ a rj o + e j, (x j ) x j n r x x r x j c j x = x 0 + c 1 a 1 + + c n r a n r, (c j )

(2016 2Q H) 17 Ax = o x = o Ax = o Ax = o s c 1,, c s x = c 1 a 1 + + c s a s Ax = o a 1,, a s Ax = o Ax = o V = {x Ax = o} = {c 1 a 1 + + c s a s c 1,, c s } Ax = o Ax = c x = d d Ax = c A(x d) = o x d Ay = o x = d + y Ax = c () + ( Ax = o ) 1 0 a d 1 0 1 b d 2 0 0 0 0 x 1 + ax 3 = d 1 x 2 + bx 3 = d 2 0 = 0 x 1 = d 1 ax 3 x 2 = d 2 bx 3 x 3 = x 3 c = x 3 x 1 x 2 = d 1 d 2 + c a b, (c ) x 3 0 1 (152) ( ) x n Ax = c (1) rank(a c) = rank A = n (2) rank(a c) = rank A < n n rank A (3) rank(a c) = rank A + 1

18 (2016 2Q H) [ ] AX = B (A B) AX = B X A, B (A B) (A B ) = O 1 A 11 0 A 12 0 0 A 1r b (1) 0 O 1 A 22 0 0 A 2r b (2) 0 O 1 0 A 3r b (3) 0 1 A rr b (r) O O O B 1 (b (i) B 1 ) B 1 = O AX = B AX = B A X = B A X = B X AX = B X, B X = (x 1 x n ), B = (b 1 b n) A x j = b j x j = y j + s c kj a k k=1 X = (x 1 x n ) = (y 1 y n ) + (a 1 a s )(c kj ) k,j Y = (y 1 y n ) AX = B 1 a i Aa i = o I ij (a 1 a s )I ij AX = B Ax = o s AX = B ns XA = B t A t X = t B ( t A t B) t X XA = B X

(2016 2Q H) 19 [ ] A n A A A A n A I n P 1,, P k P k P 1 A = I n P 1 k A = P1 1 P 1 k,, P 1 1 I n = P 1 1 P 1 k AP k P 1 = (P1 1 P 1 )(P k P 1 ) = I n P = P k P 1 P A = I n AP = I n A A 1 = P A A I n A P k P k 1 P 2 P 1 I n = P k P k 1 P 2 P 1 = P = A 1 A A 1 (A I n ) A I n P k P k 1 P 2 P 1 (A I n ) = P k P k 1 P 2 (P 1 A P 1 ) = (P k P k 1 P 2 P 1 A P k P k 1 P 2 P 1 ) = (P A P ) = (I n A 1 ) A A 1 (A I n ) P P (A I n ) = (P A P ) = (I n P ) P A = I n A P = A 1 A 1 (A I n ) 0 P P A = P A P O A A I n (A I n ) (I n X) X = A 1 (A I n ) 0 A k

20 (2016 2Q H) ( ) A n I n n n 2n (A I n ) (1) (A I n ) (I n X) A A 1 X (2) (A I n ) (A X) A 0 A n 1 A 1 1/c (2) 1 a, b x, y ax + by = 1 a, b d x, y ax + by = d a 11 A = ai1 = 1 a n1 a 11 1 r 1 r i 1 a 11 r j a j1 r 1 (j 2) (a i1 =a 11,a j1 =a j1,j i) 1 0 0 a i1, a j1 x, y xa i1 + ya j1 = 1 x = y = 0 y 0 (x 0 ) a i1 a j1 yr j a i1 ya j1 r j +xr i 1 a 11,, a n1 d 1 d a i1

(2016 2Q H) 21 [ ] A 0 {}}{ 0 {}}{ 0 {}}{ 0 {}}{ O 1 A 11 0 A 12 0 0 A 1r 0 O 1 A 22 0 0 A 2r 0 O 1 0 A 3r 0 1 A rr O O O A rank A ( ) 1 2 r 1 O 1 1 = Ir O O O O O A A A r A ( ) Ir O A A O O r A m n A P, Q P, Q Ir O P AQ = O O r = rank A P, Q P AQ Ir O O O Ir O P, Q P AQ = rank A = r O O

22 (2016 2Q H) A n rank A = n A n A A n A n A In = A O n A = I n n n P = P k P 1 P A = I n A A, X n AX = I n P P A = B P = P I n = P AX = BX P P B B n B = I n P = X XA = P A = B = I n AX = I n XA = I n A X = A 1 X 1 = A XA = I n X = A 1 A n AX = I n XA = I n X A X = A 1 ( ) n A (1) A (2) A I n (3) rank A = n (4) XA = I n AX = I n n X (4) n X A A = P 1 1 P 1 k P 1 k,, P 1 1 ( ) A m n B n l P 1,, P k (1) 0 rank A min{m, n} rank A = 0 A = O (2) rank( t A) = rank A, rank(a) = rank A (3) P, Q rank(p A) = rank(aq) = rank(p AQ) = rank A (4) rank(ab) min{rank A, rank B}

(2016 2Q H) 23 [ ] n {1, 2,, n} S n S n σ S n p i {1,, n} i j p i p j σ = ( p 1 p 2 p i p n ) σ i p i σ(i) σ S n i < j σ(i) > σ(j) σ(i) σ(j) {σ(i), σ(j)} t(σ) t(σ) σ = ( p 1 p 2 p i p n ) t(σ) = n (p i p i ) i=1 σ S n σ ε(σ) ε(σ) = ( 1) t(σ) n A = (a ij ) A A A = σ S n ε(σ)a 1σ(1) a 2σ(2) a nσ(n) σ S n n! S n σ n = 2 {1, 2} σ 1 = (1 2) σ 2 = (2 1) 0, 1 ε(σ 1 ) = 1, ε(σ 2 ) = 1 n = 3 A = ε(σ 1 )a 11 a 12 + ε(σ 2 )a 12 a 21 = a 11 a 22 a 12 a 21 S 3 = {(1 2 3), (2 3 1), (3 1 2), (1 3 2), (2 1 3), (3 2 1)} 1 1 A = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 11 a 23 a 32 a 12 a 21 a 33 a 13 a 22 a 31 n = 2, 3 () n 4

24 (2016 2Q H) [ ] ( 0 ) (I) 0 0 a 11 a 12 a 1n 0 0 0 a n1 a n2 a nn = a 11 0 a 1n a 21 0 a 2n a n1 0 a nn = 0 (II) 1 1 (1, 1) 0 a 11 a 12 a 1n 0 a 22 a 2n 0 a n2 a nn = a 11 0 0 a 21 a 22 a 2n a n1 a n2 a nn = a 11 a 22 a 2n a n2 a nn (1) a 1i a 1j a ni a nj = a 1j a 1i a nj a ni (2) c c a 11 ca 1i a 1n a n1 ca ni a nn = c a 11 a 1i a 1n a n1 a ni a nn (3) c a 1i a 1j a ni a nj = a 1i a 1j + ca 1i a ni a nj + ca ni i j j i j 0 0 (I) (II)

[] (2016 2Q H) 25 (1) t A = A (2) A = A (3) AB = A B (4) A + B = A + B ( ) a 1 a i 1 a i + a i a i+1 a n = a 1 a i a n + a 1 a i a n a j A B (5) X X = C D X = AD CB, X = A D B C A, B, C, D n c (1) A B O D = A D A C O ( t ) ( D = t A t C t ) O t = A t C D O t = t A t D = A D D (2) 1 i n A B C D i n + i 1 A B C D = C D ( 1)n A B (3) 1 i n A B C D i c c c ca cb C D = A B cn C D (4) 1 i n A B C D i c n + i A B C D = A B C + ca D + cb (5) X n (1) I n X O = 1 A C A C B D = I n O B D = A C X I n A C B D I n O I n B D = A + XC B + XD C D X = A AX + B C CX + D I n X

26 (2016 2Q H) [ ] n A A A 1 AA 1 = I AA 1 = A A 1 = I = 1 A = 0 A rank A < n P P A = P A = 0 P P = 0 A = 0 A A = 0 ( ) n A (1) A (2) AX = XA = I n X (3) AX = I n XA = I n X (4) A I n (5) A I n (6) A I n (7) rank A = n (8) Ax = o (9) Ax = c (10) A = 0 n A = (a ij ) A tr A tr A = n a ii = a 11 + a 22 + + a nn i=1 A = (a ij ), B = (b ij ) n AB (i, i) n n n n tr(ab) = a ij b ji = b ji a ij = tr(ba) i=1 j=1 j=1 i=1 n a ij b ji j=1 ( ) A, B, C n P n c (1) tr (A + B) = tr A + tr B (2) tr (ca) = c tr A (3) tr (AB) = tr (BA) (4) tr (ABC) = tr (BCA) (5) tr (P 1 AP ) = tr (A)

(2016 2Q H) 27 [ ] 1 1 (1, 1) 0 a 11 a 12 a 1n a 11 0 0 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 i j 0 A 11 o A 12 a ij A 21 o A 22 = A 11 A 12 o a ij o A 21 A 22 = ( 1)i+j a ij A 11 A 12 A 22 A 22 A = (a ij ) n A a ij ã ij A i j n 1 ( 1) i+j ( ) n A = (a ij ) (i, j) ã ij a 11 a 1 j 1 a 1 j+1 a 1n ã ij = ( 1) i+j a i 1 1 a i 1 j 1 a i 1 j+1 a i 1 n a i+1 1 a i+1 j 1 a i+1 j+1 a i+1 n a n1 a n j 1 a n j+1 a nn A = (a ij ) i a ij 0 A = a ij ã ij A = (a ij ) j a ij 0 A = a ij ã ij

28 (2016 2Q H) A 1 a + b A 2 = A 1 a A 2 + A 1 b A 2 A = (a ij ) i a (i) = a i1 t e 1 + + a in t e n A = a i1 ã i1 + + a in ã in ( ) A = (a ij ) (i, j) ã ij (i, j) A = A = n a ik ã ik = k=1 n a kj ã kj k=1 n a ik ã ik i A = k=1 j 1 1 A = a 11 ã 11 + a 12 ã 12 + + a 1n ã 1n n a kj ã kj k=1 () = a 11 ã 11 + a 21 ã 21 + + a n1 ã n1 () n A (i, j) A (j, i) A ( ) A Ã ( ) ã 11 ã n1 Ã = ã 1n ã nn AÃ = ÃA = A I n A = 0 A A 1 = 1 A Ã

[ ] n n n A = (a 1 a 2 a n ), (2016 2Q H) 29 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 a i = a 1i a 2i a ni b = b 1 b 2 b n, x = Ax = b A A A 1 x = A 1 b A 1 A 1 = 1 A Ã = 1 A (ã ji) ij x 1 x 2 x n x x = 1 A (ã ji) ij (b j ) j = 1 A n ã ji b j j=1 i A i b n a 1 b i a n = b k ã ki k=1 x i = 1 A n j=1 ã ji b j = 1 A a 1 b i a n A ( ) A = (a 1 a 2 a n ) n Ax = b x i = a 1 a i 1 b a i+1 a n a 1 a i 1 a i a i+1 a n

30 (2016 2Q H) [] x, y 2 x, y A = (x y) x, y 2 S S = det A x y det A > 0 x y det A < 0 x, y 2 det A x, y, z 3 x, y, z A = (x y z) 3 x = (x, y) x y x 1 x 2 x 3, y = y 1 y 2 y 3 (x, y) = x 1 y 1 + x 2 y 2 + x 3 y 3, x y = x 2y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 (x, x y) = x 1 (x 2 y 3 x 3 y 2 ) + x 2 (x 3 y 1 x 1 y 3 ) + x 3 (x 1 y 2 x 2 y 1 ) x = x 2 y 2 1 x 3 y 3 x 2 x 1 y 1 x 3 y 3 + x 3 x 1 y 1 x 2 y 2 = x 1 x 1 y 1 x 2 x 2 y 2 x 3 x 3 y 3 = 0 (y, x y) = 0 x y x, y x y z x y z = (x, y z) x, y, z () det A (x, y, z) det A > 0, det A < 0 2 A = (a 1, a 2 ), B = (b 1, b 2 ) A, B (a 2 b 2 )(x b 1 ) (a 1 b 1 )(y b 2 ) = 0 (a 2 b 2 )(x b 1 ) (a 1 b 1 )(y b 2 ) a 1 b 1 x = (a 2 b 2 )x (a 1 b 1 )y + (a 1 b 2 a 2 b 1 ) = a 2 b 2 y 1 1 1 A, B a 1 b 1 x a 2 b 2 y 1 1 1 = 0

[ ] V = R n a 1,, a r V (2016 2Q H) 31 c 1,, c r R c 1 a 1 + c 2 a 2 + + c r a r = o c 1 = c 2 = = c r = 0 a 1,, a r 1 a 1,, a r n r A = (a 1 a r ) a 1,, a r 1 Ax = o rank A = r a 1,, a n V 1 c V c 1,, c n R c = c 1 a 1 + c 2 a 2 + + c n a n a 1,, a n V A = (a 1 a n ) a 1,, a n V a 1,, a n 1 c V Ax = c rank A = n e 1,, e n R n R n R n V = R n ( ) a 1,, a n x V x 1,, x n R x = x 1 a 1 + x 2 a 2 + + x n a n = (a 1 a n ) x 1 x n x x 1 x n ( ) (a1,, a n )

32 (2016 2Q H) V = R n, W = R m f : V W (1) x, y V f(x + y) = f(x) + f(y) (2) x V c R f(cx) = cf(x) f : V W W ( ) b 1,, b m x V (a 1,, a n ) x 1 x = x 1 a 1 + x 2 a 2 + + x n a n = (a 1 a n ) x n f(x) W (b 1,, b m ) f(x) = y 1 b 1 + y 2 b 2 + + y m b m = (b 1 b m ) y 1 y m m n A y 1 x 1 = A y m x n A f V (a 1,, a n ) W (b 1,, b m ) x = f(s, t), y = g(s, t) s, t C 1 s, t s, t x, y x, y x = f(s + s, t + t) f(s, t) = f(s + s, t + t) f(s, t + t) + f(s, t + t) f(s, t) = f f (s, t + t) s + s t (s, t) t + ε 1 = f f (s, t) s + s t (s, t) t + ε 1 y = g g (s, t) s + s t (s, t) t + ε 2 (ε s, t 2 ) 1 x x x = s t s y y y t s t (x, y) x/ s x/ t (s, t) = det y/ s y/ t ()