線形空間の入門編 Part3
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- きみつぐ ひろき
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1 Part3 j1701 March 15, 2013 (j1701) Part3 March 15, / 46
2 table of contents (j1701) Part3 March 15, / 46
3 f : R 2 R 2 ( ) x f = y ( ) ( x y ) = ( ) x y y x, y = x ( x y) 0!! ( ) ( ) ( ) f = = f 1 ) ( 2 2 = 1 1 ( ) = ( 0 0 ) (j1701) Part3 March 15, / 46
4 (j1701) Part3 March 15, / 46
5 y = x?? y = x,,!! ( ) x f = y f 0,!! f 0 ( ) x y = y x ( ) 0 0 (j1701) Part3 March 15, / 46
6 f : V W, Ker f def = {x V f(x) = 0} V Ker f f (Kernel) V W f 0 Ker f (j1701) Part3 March 15, / 46
7 Ker f Ker f!! (j1701) Part3 March 15, / 46
8 Ḳer f, V 3!! Ker f v, w Ker f v + w Ker f v Ker f, c R cv Ker f (j1701) Part3 March 15, / 46
9 ,, 0 Ker f f(0) = 0 v, w Ker f v + w Ker f f(v + w) = f(v) + f(w) = = 0, v + w Ker f c R, v Ker f cv Ker f f(cv) = cf(v) = c 0 = 0, cv Ker f (j1701) Part3 March 15, / 46
10 f : R 2 R 2 ( ) x f = y ( ) ( x y ) =, 4 f ( ) ( ) ( ) ( ) ,,, ( ) x y y x (j1701) Part3 March 15, / 46
11 ( ) 1 f 2 ( ) 4 f 1 ( ) 1 f 1 ( ) 3 f 2 = = = = ( ) ( ) ( ) = ( ) ( ) ( ) = ( ) ( ) ( ) = ( ) ( ) ( ) = (j1701) Part3 March 15, / 46
12 y = x?? (j1701) Part3 March 15, / 46
13 ( x y) R 2 ( ) ( x y ) = ( ) ( ) ( ) x y 1 1 = (x y) y x 1 1 (, R 2, 1 ) 1!!!! (j1701) Part3 March 15, / 46
14 f : V W, Im f def = {f(x) W x V } W Im f f (Image) V f W Imf (j1701) Part3 March 15, / 46
15 Ịm f W Proof f(0) = 0, 0 Im f f(v), f(w) Im f, c R, f(v) Im f,, Im f W f(v) + f(w) = f(v + w) Im f cf(v) = f(cv) Im f (j1701) Part3 March 15, / 46
16 ( ) 1 Ker f = 1 ( ) 1 Im f = 1 dim (Im f) + dim (Ker f) = 2 = dim R 2 ( (dimension theorem)!!) (j1701) Part3 March 15, / 46
17 f f : R 3 R 3 x x f y = y z z!! x x 0 Ker f = y f y = 0 z z 0 Im f = x x f y y R 3 z z (j1701) Part3 March 15, / 46
18 , Ker f x y = z 0 OK (j1701) Part3 March 15, / 46
19 , { x 3z = 0 x = 3z y + z = 0 y = z z = t, x = 3t, y = t, x 3t 3 y = t = t 1 (t R) z t 1 3 Ker f = 1 dim (Ker f) = 1 1 (j1701) Part3 March 15, / 46
20 Im f x y R 3, z x y z x x + 2y z y = y + z = x 0 + y 1 + z z x + y 2z (j1701) Part3 March 15, / 46
21 , 1, , Im f!! (why?) Im f,!! (j1701) Part3 March 15, / 46
22 3 3, , ,!! (j1701) Part3 March 15, / 46
23 , = , x f y = x 0 + y 1 + z z , x 1 2 f y = (x 3z) 0 + (y + z) 1 z 1 1 (j1701) Part3 March 15, / 46
24 x 3z = t, y + z = s, x 1 2 f y = t 0 + s 1 z , 1, Im f!! , 1 Im f, dim (Im f) = (j1701) Part3 March 15, / 46
25 , ( ) ( ),, (j1701) Part3 March 15, / 46
26 !! V, W : linear space, W V V/W (quotient linear space)!! (j1701) Part3 March 15, / 46
27 1 a = a ( ) 2 a = b b = a (a b b a ) 3 a = b, b = c a = c ( ), (j1701) Part3 March 15, / 46
28 a, b 3,, a b mod 3 1 a a mod 3 ( 3 ) 2 a b mod 3 b a mod 3 (a b, b a ) 3 a b mod 3, b c mod 3 a c mod 3 (a b, b c, a c ) a b mod 3, a, b 3 (j1701) Part3 March 15, / 46
29 P, Q,, P Q 1 P Q ( ) 2 P Q Q P (P Q, Q P ) 3 P Q, Q R P R (P Q, Q R, P R ) (j1701) Part3 March 15, / 46
30 ,, (equivalent relation) 1 a a ( / Reflexive) 2 a b b a ( / Symmetric) 3 a b, b c a c ( / Transitive) (j1701) Part3 March 15, / 46
31 (j1701) Part3 March 15, / 46
32 1 3 0 def = def = def = 2 0, 1, 2, 0, 1, 2 Z 0 / def = { 0, 1, 2} Z 0 (quotient set) (j1701) Part3 March 15, / 46
33 !! X, X/ def = { x x X}, X (quotient set) ( x = {y X x y}) (j1701) Part3 March 15, / 46
34 V R-, W v w def v w W, V, V V/W def = V/, V/W V W (quotient linear space) (j1701) Part3 March 15, / 46
35 V/W = { v v V } R- x + ȳ = x + y c x = cx (j1701) Part3 March 15, / 46
36 well-defined ( ) v, w v, w v + w = v + w v v, w w, x, y W st v + x, w = w + y v + w = v + w = v + x + w + y = v + w + x + y = v + w + 0 = v + w (j1701) Part3 March 15, / 46
37 c v = c v v v, x W st v = v + x c v = cv = c(v + x) = cv + cx = cv + cx = cv + 0 = cv well-defined, V/W, 8, R- (j1701) Part3 March 15, / 46
38 f : V W, f (linear isomorphism) f f : V W, V, W, V, W (isomorphic), V W ( ) (j1701) Part3 March 15, / 46
39 ver f : V W f φ : V/Ker f Im f 4!! φ( v) = f(v) 1 φ well-defined ( ) 2 φ 3 φ 4 φ (j1701) Part3 March 15, / 46
40 φ well-defined v V/Ker f, v v v, x Ker f st v = v + x φ( v ) = f(v ) = f(v + x) = f(v) + f(x) = f(v) + 0 ( x Ker f), f(v ) = f(v), well-defined (j1701) Part3 March 15, / 46
41 φ φ(v + w) = f(v + w) = f(v) + f(w) = φ( v) + φ( w) φ(cv) = f(cv) = cf(v) = cφ( v), φ (j1701) Part3 March 15, / 46
42 φ f(v) Im f v V/Ker f, φ( v) = f(v), φ, f Ker f = {0} (!!) v Ker φ, φ( v) = f(v) = 0, v Ker f v = 0, Ker φ = { 0}, φ (j1701) Part3 March 15, / 46
43 V R-, W, dim V/W = dim V dim W Proof ( ) (j1701) Part3 March 15, / 46
44 (dimension theorem) f : V W dim (Im f) + dim (Ker f) = dim V Proof, dim, V/Ker f Im f, dim V dim (Ker f) = dim (Im f) (j1701) Part3 March 15, / 46
45 1, 2 3 ( ver) (j1701) Part3 March 15, / 46
46 Thank you!!! Good luck!!! (j1701) Part3 March 15, / 46
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