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1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml

2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128 ( 19 13 21 131 ( 21 132 23 14 24 141 24 142 24 143 25 15 27

3 1 11 111 y = f(x = ax (11 x y R R := {x : < x < } (11 R R z = f(x, y = ax + by (12 w = f(x, y, z = ax + by + cz (13 a, b, c x, y, z w n n a 1,, a n,, y y = f(,, = a 1 + + a n (14 n 1

4 1 112 (14 a 1 + + a n = (a 1 a 2 a n (15 o (a 1 a 2 a n 111 (12 (a 1,, a n y = f(,, = a 1 + + a n n y,, 113 n 112 (a 1 a 2 a n n n n R n n n n n R n 113 R 2 xy R 3 3 (xyz 114 R n n n google R n n

11 5 114 111 y = ax y = ax, z = bx (16 x y, z a, b y a = x (17 z b (16 n y 1, y 2,, y n x a 1, a 2,, a n y 1 = a 1 x,, y n = a n x y 2 x y 1 y 1 y n a 1 y 2 = a 2 x y n a n 115 n,,, m y 1, y 2,, y m y 1,, 111 y 1 = (a 1 a 2 a n y 2,, 111 y 2 = (a 1a 2 a n

6 1 a 1 a 1 y 3, y 4 a 1, a 1 a y 1 = (a 11 a 12 a 1n y 2 = (a 21 a 22 a 2n y m = (a m1 a m2 a mn n m a ij m n y 1 y 2 a 11 a 12 a 1n = a 21 a 22 a 2n (18 y m a m1 a m2 a mn m n m n m n n m 11 C

12 7 12 121 121 x y z w z = x + y w = 2x + 4y (15 x z = (1 1 y x w = (2 4 y (18 z 1 1 x = w 2 4 y x = 7, y = 3 ( 1 1 7 1 7 + 1 3 10 = = 2 4 3 2 7 + 4 3 26 ( a ( a c b x ax + by = c d y cx + dy b x d y (19 (110 122 x y z u v u = x + y + z v = 2x + 4y + 6z (111 (18 x u 1 1 1 = y (112 v 2 4 6 z ( x + y + z 2 3 3 2x + 4y + 6z 2 m n n m

8 1 := A := a b c e f g A a b c e f g 2 3 (a b c A (e f g A a b c,, e f g A i j (i, j m n 1 m (i, j x x x a b c A :=, x := y e f g z x ( a b c ax + by + cz Ax = y = e f g ex + fy + gz z x 123 ( x = y θ u ( (cos θx (sin θy u = (sin θx + (cos θy u = ( cos θ sin θ sin θ x cos θ 2 2 θ 124 θ ( cos θ sin θ sin θ cos θ θ R(θ radian

12 9 122 A = a b p q, P = c d r s 2 2 x P x A A(P x 125 S, T S ( s S T S T (mapping (function f : S T s T f(s s f(s s f(s f S T f : S T, S f T diagram f : S T s f(s S f (domain T f (codomain 126 f(x = + 1 R R x + 1 127 x R 2 2 2 A Ax f A : R 2 R 2 x Ax P f P : R 2 R 2, f P (x = P x f A P 128 f : S T, g : T U g f : S U s S (g f(s = g(f(s

10 1 A(P x = f A (f P (x = f A f P (x A(P x P x = px + qy rx + sy A(P x = ( a c ( b px + qy a(px + qy + b(rx + sy = d rx + sy c(px + qy + d(rx + sy ( (ap + brx + (aq + bsy = (cp + drx + (cq + dsy ( ap + br cp + dr aq + bs x cq + ds y a b p q 129 A =, P = A B AB c d r s AB = ( ap + br cp + dr aq + bs cq + ds A(P x = (AP x AP p 1210 AP A r A P AP A P A 2 2 P 2 3 x 3 P x 2 A(P x 2 AP A(P x = (AP x AP 2 3 1211 A l m P m n l n C A(P x = Cx n x C C A P AP

12 11 123 x x = θ u y x (cos θx 0 0 (sin θx y (sin θy u (cos θy (cos θx (sin θy u = + = (sin θx (cos θy ( cos θ sin θ sin θ x (124 u = R(θx cos θ y 1212 ( cos α cos β sin α sin β cos α sin β + sin α cos β R(αR(β = R(α + β ( cos α sin β sin α cos β cos(α + β sin(α + β = cos α cos β sin α sin β sin(α + β cos(α + β R(βx x β R(α(R(βx α (R(αR(βx R(α + βx x R(αR(β = R(α + β Ax = Bx x A = B x i A B i 1213 (1212 sin, cos 124 1211 AP A(P x = (AP x x AP

12 1 1214 A l m P m n P = (p 1 p 2 p n p i P i m AP = A (p 1 p 2 p n = (Ap 1 Ap 2 Ap n AP i Ap i 12 125 R n n (n R m m (m x, x R n x + x R n λ R x λ λx λ n 1215 f : R n R m 17 x, x R n, λ R 1 f(x + x = f(x + f(x 2 f(λx = λx 1216 1 2 (1 (2 1217 e 1 R n 1 1 0 e 2 R n 2 1 0 ( e n R n n 1 0 e i (e 1,, e n 1218 f : R n R m p 1 := f(e 1, p 2 := f(e 2, p n := f(e n, P := (p 1 p 2 p n P m n f(x = P x x R n

12 13 13 x = e 1 + + e n f f(x = f(e 1 + + f(e n P x P f P f P f(x := P x 14 1214 (AP (AP e 1 = A(P e 1 = Ap 1 A, B m n A + B (A + Bx = Ax + Bx 0 m n O A + O = A = O + A I n A = A I n n n (i, i i 1 0 n n (unit matrix I n E n I, E A n m I n A = A, B l n BI n = B (ABC = A(BC (f g h = f (g h 126 2 2 A n P A = AP = I n n P P A A 1 15 QA = AQ = I n Q = P a b A = c d d b det A := ad bc 0 1/(det A c a

14 1 16 1219 3 3 det A 0 A 1 = 1/(det AÃ Ã 1220 R(θ := ( cos θ ( cos θ sin θ sin θ cos θ cos θ cos θ ( sin θ sin θ = 1 sin θ sin θ R(θ A AR(θ = I 2 x R(θ A θ A A R( θ R(θ 1 = R( θ = cos θ ( cos( θ sin( θ sin( θ cos( θ cos( θ = cos θ, sin( θ = sin θ 1221 (110 z 1 1 x = w 2 4 y x, y (z, w 2 2 A A 1 z = Ax A 1 z = A 1 (Ax = (A 1 Ax = I 2 x = x x = A 1 z 2 2 A 1 4 1 2 1/2 = 1/2 = 2 1 1 1/2 10 (10, 26 z = A 1 7 x = 26 3

12 15 127 41 4, 5 122 x y z u v w 4 u = x + y + z v = 2x + 4y + 6z w = 2x + 0y + 4z (113 u 1 1 1 x v = 2 4 6 y (114 w 2 0 4 z 8 1 1 1 x 24 = 2 4 6 y (115 14 2 0 4 3 3 (113 x 2 (114 1 0 0 2 1 0 8 1 1 1 x 8 = 0 2 4 y (116 14 2 0 4 ( 2 x (3, 1 ( 2 3 1 0 0 0 1 0 2 0 1 z z

16 1 8 1 1 1 x 8 = 0 2 4 y (117 2 0 2 2 ( 2 (113 z 8 = x + y + z 8 = 0x + 2y + 4z 2 = 0x 2y + 2z (118 (= x (=(2, 1, (3, 1 0 x y y 1/2 y 1 1 0 0 0 1/2 0 8 1 1 1 x 4 = 0 1 2 y (119 2 0 2 2 y y 1 1 1 0 0 1 0 4 1 0 1 x 4 = 0 1 2 y (120 2 0 2 2 2 1 0 0 0 1 0 0 2 1 4 1 0 1 x 4 = 0 1 2 y (121 6 0 0 6 z z z

12 17 y z 1/6 1 0 0 0 1 0 /6 4 1 0 1 x 4 = 0 1 2 y (122 1 z z 1 0 1 0 1 0 5 1 0 0 x 4 = 0 1 2 y (123 1 z z 1 0 0 0 1 2 5 1 0 0 x 2 = 0 1 0 y (124 1 z z x y x = 5, y = 2, z = 1 z P41 P ij (c 1 P i (c 1 0 1 1 1 0 1 0 0 0 1 2 0 1 0 0 1 0 0 1 0 0 1 0 0 1/2 0 /6 0 2 1 1 0 0 u 1 0 0 x 0 1 0 2 1 0 v = 0 1 0 y 2 0 1 w z

18 1 1 1 1 2 4 6 2 0 4 1 1 1 1 1 1 0 2 4 = 2 1 0 2 4 6 2 0 4 2 0 4 1 1 1 1 1 1 1 1 1 0 2 4 = 0 1 0 0 2 4 = 0 1 0 2 1 0 2 4 6 0 2 2 2 0 1 2 0 4 2 0 1 2 0 4 1 0 1 1 1 0 1 0 0 0 1 0 = 0 1 2 0 1 0 0 1 0 0 1 0 0 1 0 0 1/2 0 P /6 I 3 = P A 0 2 1 1 1 1 0 1 0 2 1 0 2 4 6 2 0 1 P A (Gaussian ellimination A P I 3 = P A = AP AP = I 3 (invertible matrix (regular matrix 1222 n A, P I n = P A P P 1 I n = AP A P I n = P A P P = I n P = P AP P 1 I n = P 1 P = P 1 P AP = I n AP = AP P 1223 P, Q n P Q Q 1 P 1 1224 42 2 0 4

12 19 128 ( 1225 n A n m m = 1 ( P ij (c I n (i, j c c i j P ij (ca A j c i P ij (c P ij ( c ( P i (c I n (i, i c c 0 P i (c P i (ca A i c P i (c P i (c 1 ( P ij I n (i, i (j, j 0 (i, j (j, i 1 P i (ca A i j P i (c P i (c 1 1 0 0 P 21 ( 2 = 2 1 0 1 0 0 P 31 ( 2 = 0 1 0 2 0 1 1 0 0 P 2 (1/2 = 0 1/2 0 1 1 1 A = 2 4 6 P 21 ( 2A 2 0 4 (2, 1 0 P 31 ( 2P 21 ( 2A (3, 1 0 x P 32 (2P 12 ( 1P 2 (1/2P 31 ( 2P 21 ( 2A

20 1 (1, 2, (3, 2 0 y P 23 ( 2P 13 (1P 3 (1/6P 32 (2P 12 ( 1P 2 (1/2P 31 ( 2P 21 ( 2A z P I n = P A P = A 1 P (I 3 A 3 6 (A P (I 3 A = (P P A = (P I 3 P 0 A = (1, 1 1 2 1 3 P 12, P 13 (1, 1 0 0 1226 n A i 0 A I n = P A P P A P 0 (rank 1227 A n P P = A 1 A A (1, 1 A 0 A (1, 1 1 1 (i, 1 0 (i = 2, 3,, n (2, 2 i (i > 2

13 21 (2, i i = 2, 3,, n 0 (2, 2 1 2 (i, 2 (i 2 0 A P 1, P 2,, P k P k P 1 A = I n P = P k P 1 P 1 = A P = P k P 1 P 1 = P1 1 P 1 k 13 131 ( 131 ( S, T S T S T := {(s, t s S, t T } S S S 2 (S S S S 3 132 R 2 xy R 3 xyz 133 f : S S S S f(s 1, s 2 s 1 s 2 g : S S S T S S T S 134 S = R (s 1, s 2 s 1 + s 2 R x R x R S = R n + S x x R n R R n R n, (λ, x λx ( λx λ x λ R R n 135 ( V + V (V, + V + V

22 1 1 x, y, z V (x + y + z = x + (y + z + (associativity law (V, + (semi group 2 0 V x V x + 0 = x = 0 + x(, identity law 0 + (V, +, 0 3 g : V V x V x + g(x = 0 = g(x + x g(x x (inverse element g(x x (V, +, 0, 4 x + y = y + x (V, +, 0, 136 ( (V, +, 0, R V R V V, (λ, x λ x λ x λx 5 λ (x + y = λ x + λ y 6 (λ + R µ x = λ x + V µ x 7 (λ R µ x = λ (µ x 8 1 x = x (V, +, 0,, R R n 137 R n V f, g V f + V g x f(x + R g(x

13 23 132 138 V, W V W (linear map V W f 1 f(x + V y = f(x + W f(y 2 f(λ V x = λ W f(x λ f f(0 = 0, f( x = f(x f(0 = f(0 + 0 = f(0 + f(0 f(0 f( x + f(x = f(0 = 0 f(x 139 1310 A m n x Ax A ( := f : R n R m R n R m m n A A ( ( 1218 1311 V F (t V F (t F (t V V F (t 1 F (tdt V R 0 1312 V, W f : V W f g f 17 g 1313 f : R n R n n A f A n P, Q x P x = Qx P = Q x g B g BAx = x, ABx = x x BA = I n = AB A A 1

24 1 14 141 141 142 V n v 1,, v n V,, R v 1 + + v n V v 1,, v n,, 143 φ v φ v : R n V = v 1 + + v n v n φ v1,,v n φ v φ v = (v 1,, v n 144 φ v : R n V f : R n V f(e i v i f = φ v 142 φ v 145 φ v : R n V v 1,, v n V (generate V V n 146 φ v : R n V v 1,, v n (= =linearly independent (= =linearly dependent

14 25 147 v 1,, v n φ v 0 {0} v 1 + + v n = 0 = = = 0 0 0 {0} φ v = φ v y 1 y 2 y n φ v φ v y 1 y 2 = y 1 y 2 y n 0 0 {0} y 1 y 2 = 0 = y 1 y 2 y n y n y n 143 148 ( φ v : R n V v 1,, v n V V V R n 149 v 1,, v n V 1 V 2 1410 V = R 2 a c, b d a c x + y = 0 b d x = y = 0 a c x 0 = b d y 0 x = y = 0 2 2 A ( A 1 x = y = 0 A a c, b d

26 1 ad bc = 0 0 a 0 c/a 0 1411 t V := {A sin(t + C A, C R} f (t = f(t V x, y x sin t + y cos t A sin(t + C V sin t, cos t V V t = 0, t = π/4 sin t, cos t V R 2 x V, x sin t + y cos t y 1412 ( V v 1, v 2,, v n W v w, w 2,, w m f : V W R n φv V f W φ 1 w R m m n A A f v, w 1413 f(v 1 w i = 2,, n f(v 1 = a 11 w 1 + a 21 w 2 + + a m1 w m f(v i = a 1i w 1 + a 2i w 2 + + a mi w m A := (a ij A a ij x R n φ v (x = (v 1,, v n

15 27 f f A f(φ v (x = (f(v 1,, f(v n (f(v 1,, f(v n = (w 1,, w m A x f(φ v (x = (f(v 1,, f(v n x = (w 1,, w m Ax = φ w (Ax φ 1 w f φ v (x = Ax 1414 V, W Jordan 15 V n V 8 151 V v 1,, v m V v 1,, v m V < v 1,, v m > v 1,, v m V 152 V v 1,, v m v 1,, v m, w w < v 1,, v m > v 1 + + x m v m + yw = 0 0,, x m, y y = 0 v 1,, v m y 0 yw < v 1,, v m > y 153 V {v 1,, v m } V V

28 1 154 V T = {v 1,, v m } V m S S S T (T S = T S V 155 S T S S V T w < S > w S w / S w < S > T < S > < T > < S > V =< T > V =< T > < S > V V =< T > 156 ( rank v 1,, v m V rank {0} 0 157 ( a 1,, a n n b 1,, b m < a 1,, a n > b 1,, b m m n a 1,, a n < a 1,, a n >=< a 1,, a n > b 1,, b m b 1 a 1,, a n b 1 0 0 0 a 1 b 1 = a 1 + + a n, 0 a 1 < b 1, a 2,, a n > < a 1,, a n > < b 1, a 2,, a n > V b 1, a 2,, a n a 2,, a n b 2 b 2 b 1, a 2,, a n a 2,, a n 0 b 1, b 2 0 a 2 < b 1, a 2,, a n >=< b 1, b 2, a 3,, a n > b 1,, b m a 1,, a m n m n < m a b n b b n +1 < b 1, b 2,, b n >

15 29 158 V n V n V n V V V dim V V n v 1,, v n V =< v 1,, v n > V n n w 1,, w n V < w 1,, w n > v V w 1,, w n, v V =< w 1,, w n > 18 V n 1 v 1,, v n V 2 v 1,, v n V 3 v 1,, v n V V 3 2 3 v 1,, v n 145, 146 159 A m n f : R n R m A 1 f A n 2 f A n R m 1510 A n 1 A 2 A n 3 A n R m 4 A n R m 1 f B AB = I n, BA = I n 1 2, 1 3 2,3,4 2 3 2 f 1 3 2 3 1 1511 A n AX = I n n XA = I n X = A 1 XA = I n AX = I n A XA = I n A (

30 1 [1] [2]

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