γραμμική αλγεβρα ιι

Click here to load reader

  • date post

    18-Aug-2015
  • Category

    Documents

  • view

    38
  • download

    0

Embed Size (px)

Transcript of γραμμική αλγεβρα ιι

  1. 1.
  2. 2. iii
  3. 3. iii Cayley-Hamilton Jordan Gram - Schmidt R v
  4. 4. vi L(V ) C
  5. 5. (x) = anxn + an1xn1 + a1x + a0 n ai, i = 0, 1, . . . , n aixi F F R C F F[x] x (x) = anxn+an1xn1+ a1x+a0 (x) = bmxm + bm1xm1 + b1x + b0 n = m ai = bi i = 0, 1, . . . , n 0xi
  6. 6. n an (x) = anxn + an1xn1 + a1x + a0 an = 0 n deg( (x) ) = n anxn an 1 (x) = anxn +an1xn1 + a1x+a0 a1 n (x) (x) (x) = a0 0 F F[x] F[x] (x) = anxn + an1xn1 + a1x + a0 (x) = bxm + bxm1 + + b1x + b0 m n (x)+(x) = bmxm+bm1xm1+ +bn+1xn+1+(an+bn)xn+ (a1+b1)x+ (a0 + b0) (x) (x) ( + )(x) (x) = anxn+an1xn1+ a1x+a0 aixi 0xi (x) = x4+3x3+x (x) = 2x3x+1 (x) = x4 + 3x3 + 0x2 + x + 0 (x) = 0x4 + 2x3 + 0x2 x + 1 (x)+(x) = ( x4 +3x3 +0x2 +x+0 )+( 0x4 +2x3 +0x2 x+1 ) = x4 + 5x3 + 0x2 + 0x + 1 = x4 + 5x3 + 1 (x) = anxn + an1xn1 + a1x + a0 (x) = bxm + bxm1 + + b1x + b0 (x) (x) = crxr + cr1xr1 + + c1x + c0 c0 = a0b0 c1 = a0b1 + a1b0 ck = i+j=k aibj,
  7. 7. (x) (x) ( )(x) (x)(x) xixj = xi+j (x) = x4+3x3+x (x) = 2x3x+1 (x)(x) = (12)x7 +(10+32)x6 +(1(1)+30+02+10+0 0)x5 +(11+3(1)+12)x4 +(31+0(1)+01)x3 +(01+1(1)+00)x2 + (1 1 + 0 (1))x + 0 1 = 2x7 + 6x6 + (1)x5 + 0x4 + 3x3 + (1)x2 + x + 0 = 2x7 + 6x6 x5 + 3x3 x2 + x F[x] F F[x] ( (x) + (x) ) + (x) = (x) + ( (x) + (x) ) (x), (x), (x) F[x] (x) + (x) = (x) + (x) (x), (x) F[x] (x) + 0 = 0 + (x) = (x) (x) F[x] (x) = anxn + an1xn1 + a1x + a0 F[x] (x) = (an)xn + (an1)xn1 + (a1)x + (a0) F[x] (x) + ((x)) = ((x)) + (x) = 0 ( (x) (x) ) (x) = (x) ( (x) (x) ) (x), (x), (x) F[x] (x) (x) = (x) (x) (x), (x) F[x] (x) 1 = 1 (x) = (x) (x) F[x] ( (x) + (x) ) (x) = ( (x) (x) ) + ( (x) (x) ) (x), (x), (x) F[x]
  8. 8. F[x] F (x) (x) (x)+(x) = 0 deg( (x)+(x) ) max( deg (x), deg (x) ) deg( (x) (x) ) = deg (x) + deg (x) (x) F[x] (x) F[x] (x) (x) = 1 (x) (x) (x), (x) F[x] (x) (x) = 0 (x) (x) (x) R[x] ((x))11 = (x + 1)22 + (x 1)2004
  9. 9. F[x] Z Z F[x] (x), (x) F[x] (x) (x) (x) | (x) (x) F[x] (x) = (x) (x) (x) (x) (x) (x) (x) (x) (x) F[x] (x) 0 = 0 0 (x) | (x) (x) = 0 (x) | (x) (x) F[x] (x) = (x) (x) (x) F[x] (x) = (x) (x) (x) = (x) (x) = (x) (x) (x) ( (x) (x) ) = 0 (x) (x) (x) = 0 (x) = (x) (x) | (x) deg( (x) ) deg( (x) ) (x) | (x) (x) | (x) deg( (x) ) = deg( (x) ) c (x) F[x] (x) = c ( c1 (x) ) (x) | (x) 0 = c F[x] c (x) | (x) (x) = anxn + an1xn1 + a1x + a0 a1 n (x) (x) (x) (x) (x) (x) (x) (x) (x) 1(x) 2(x) (x) (x)1(x)+(x)2(x) (x), (x) F[x] p(x) F[x]
  10. 10. F F[x] F[x] c p(x) c F p(x) F[x] F p(x) = (x) (x) (x), (x) F[x] (x), (x) (x) F[x] F[x] pi(x) F[x], i = 1, 2, . . . , n c F (x) = c p1(x) p2(x) pn(x) 1 1 F x2 + 2 x2 + 2 = (x + i 2) (x i 2) (x) n (x) (x) (x) n (x) 1(x) 2(x) (x) = 1(x) 2(x) 1(x) 2(x) n (x)
  11. 11. (x), (x) F[x] (x) = 0 (x), (x) F[x] (x) = (x)(x)+(x) (x) = 0 deg((x)) < deg( (x) ) (x) (x) (x) = (x) (x) (x) (x) = 0 (x) (x) A = { (x) (x) (x) (x) F[x] } (x) = (x) (x) (x) A (x) = (x) (x) + (x) deg( (x) ) < deg( (x) ) (x) = (x) (x) (x) = anxn + an1xn1 + a1x + a0 (x) = bxm + bxm1 + + b1x + b0 deg((x) ) = n m = deg( (x)) (x) (anb1 m )xnm(x) (x) (anb1 m )xnm(x) (x) (x) (anb1 m )xnm(x) = (x) (x) (x)(anb1 m )xnm(x) = (x)( (x)+(anb1 m )xnm )(x) A (x) A deg( (x) ) < deg( (x) ) (x) (x) (x) = (x)(x) + (x) deg( ) < deg( (x) ) (x) (x) (x) (x) (x) = (x) (x) + (x) deg( (x) ) < deg( (x) ) (x) = (x)(x) + (x) (x) = (x) (x) + (x) (x)( (x) (x) ) = (x) (x) (x) (x) = 0 (x) (x) = 0 deg( (x) (x) ) deg( (x) ) deg( (x) (x) ) max( deg( (x) ), deg( (x) ) ) < deg( (x) ) (x) = (x) (x) = (x) (x) (x) (x) (x) (x) = 0 deg( (x) ) < deg( (x) ) (x) = 0 (x) = (x)
  12. 12. (x) = 4x5 3x4 7x2 +6 (x) = x3 +7x2 +3x2 (x) (x) (x) (x) (x) deg( (x) (x) (x) ) < deg( (x) ) (x) 4 (x) 4 x53 ( 4 x53 )(x) = 4x5+28x4+12x38x2 (x) (x)( 4 x53 )(x) = 31x412x3+x2+6 1(x) = 4x2 1(x) = (x)( 4 x53 )(x) = 31x4 12x3 +x2 +6 1(x) = 31x4 12x3 + x2 + 6 (x) (x) 1(x) 31 (x) 31 x43 ( 31 x43 ) (x) = 31x4 217x3 93x2 + 62x 1(x) = 31x4 12x3 + x2 + 6 1(x) ( 31 x43 ) (x) = 205x3 + 94x2 62x + 6 2(x) = (x)( 4 x2 31x )(x) = 205x3 +94x2 62x+6 2(x) = 205x3 + 94x2 62x + 6 1(x) (x) 2(x) 205 (x) 205 x33 = 205 ( 205 ) (x) = 205x3+1435x2 +615x410 2(x) = 205x3+94x262x+6 2(x)( 205 )(x) = 1341x2677x+416 (x) = 2(x)( 205 )(x) = (x)( 4x2 31x+205 )(x) = 1341x2 677x+416 (x) (x) = 4x2 31x+205 (x)
  13. 13. 4t5 3t4 + 0t3 7t2 + 0t + 6 t3 + 7t2 + 3t 2 4t5 +28t4 + 12t3 8t2 4t2 31t + 205 31t4 12t3 + t2 + 0t + 6 31t4 217t3 93t2 + 62t 205t3 + 94t2 62t + 6 205t3 + 143t2 +615t 410 1341t2 677 +416 (x), (x) F[x] c (x), (x) F[x] d(x) F[x] (x) (x) (i) d(x)| (x) d(x)| (x) d(x) (x) (x) (ii) d(x) (iii) (x) F[x] (x)| (x) (x)| (x) (x)| d(x) (x) (x) d(x) (x) (x) d(x) = ( (x), (x) ) d(x) = ( (x), (x) ) (x) (x) d1(x) d2(x) (i) (iii) d1(x)| d2(x) d2(x)| d1(x) c F[x] d1(x) = c d2(x) d1(x), d2(x) d1(x) = d2(x) (iii)
  14. 14. (x), (x) F[x] d(x) (x) (x) (x), (x) F[x] d(x) = (x) (x) + (x) (x) U = { (x)(x) + (x)(x) | (x), (x) F[x] } U (x) (x) U (x) = (x)(x) + (x)(x) U c c1(x) = (c1(x))(x) + (c1(x))(x) U d(x) = (x)(x) + (x)(x) U U d(x) (ii) (x) F[x] (x)| (x) (x)| (x) (x)| d(x) d(x) (iii) (i) (x) = (x)(x)+(x)(x) U d(x)| (x) (x), (x) F[x] (x) = (x) d(x) + (x) (x) = 0 deg( (x) ) < deg( d(x) ) (x) = (x) (x) d(x) = (x)(x)+(x)(x)(x)( (x)(x)+(x)(x) ) = ( (x)(x) (x) )(x)+ ( (x) (x) (x) )(x) U (x) c c1(x) U (x) d(x) d(x) (x) = 0 d(x) U (x) (x) (x) (x) (x)| (x) ( (x), (x) ) = c1(x) c (x)
  15. 15. (x) (x) c1, c2 F ( (x), (x) ) = ( c1 (x), c2 (x) ) (x) (x) (x) (x) ((x), (x)) = (x)(x) + (x)(x) (x) (x) (x) (x) (x) ( (x), (x) ) = ( (x), (x) ) (x) F[x] (x) = (x) (x) + (x) d1(x) = ( (x), (x) ) d2(x) = ( (x), (x) ) d1(x) (x) = (x) (x) (x) (x) d1(x)|d2(x) d2(x) (x) (x) = (x) (x) + (x) d2(x)|d1(x) d1(x) d2(x) d1(x) = d2(x) (x) (x) = 0 deg( (x) ) < deg( (x) ) (x), (x) F[x] (x) (x) = 1(x) (x) + 1(x) deg( 1(x) ) < deg( (x) ) (x) = 2(x) 1(x) + 2(x) deg( 2(x) ) < deg( 1(x) ) 1(x) = 3(x) 2(x) + 3(x) deg( 3(x) ) < deg( 2(x) ) 2(x) = 4(x) 3(x) + 4(x) deg( 4(x) ) < deg( 3(x) ) n2(x) = n(x) n1(x) + n(x) deg( n(x) ) < deg( n1(x) ) n1(x) = n+1(x) n(x) + 0 n (x) n+1(x) deg( (x) ) > deg( 1(x) ) > deg( 2(x) ) > deg( 3(x) ) >
  16. 16. ( (x), (x) ) = ( (x), 1(x) ) = ( 1(x), 2(x) ) = = ( n(x), 0 ) n(x) (x) (x) ( (x), (x) ) = (x) (x) + (x) (x) n(x) = n2(x) n(x) n1(x) n1(x) = n3(x) n1(x) n2(x) n2(x) = n4(x) n2(x) n3(x) n(x) = n3(x) n4(x) + n2(x) n3(x) n(x) = 2(x)1(x)+3(x)2(x) n(x) = 1(x)(x)+2(x)(x) r n(x) (x) = r12(x) (x) = r11(x) 2x43x33x2+2x+2 , x32x23x+4 R[x] 2x4 3x3 3x2 + 2x + 2 x3 2x2 3x + 4 2x4 3x3 3x2 + 2x + 2 = (2x + 1) (x3 2x2 3x + 4) + (5x2 3x 2) 5x2 3x2 x3 2x2 3x+4 5x2 3x2 x3 2x2 3x+4 = (1 5 x 7 25) (5x2 3x2)+(86 25 x+ 86 25) 86 25x + 86 25 5x2 3x 2 86 25x + 86 25 5x2 3x2 = (125 86 50 86 ) (86 25 x+ 86 25)+0 2x4 3x3 3x2 +2x+2 x3 2x2 3x + 4 25 86(86 25 x + 86 25) = x 1 2x4 3x3 3x2 + 2x + 2 x3 2x2 3x + 4 86 25x+ 86 25 = (x3 2x2 3x+4)(1 5x 7 25 ) (5x2 3x2) = (x3 2x2 3x+ 4) (1 5x 7 25 ) [ (2x4 3x3 3x2 + 2x + 2) (2x + 1) (x3 2x2 3x + 4) ] = (1 5 x 7 25) (2x43x33x2+2x+2)+( (1 5 x 7 25) (2x+1)+1 ) (x3 2x23x+4) = (1 5 x 7 25) (2x4 3x3 3x2 +2x+2)+(2 5x2 19 25x+ 18 25) (x3 2x2 3x+4) (86 25 )1 (1 5 x 7 25) (86 25 )1 (2 5 x2 19 25 x + 18 25 ) i(x) F[x] i = 1, 2, . . . , n d(x) F[x] i(x) i =
  17. 17. 1, 2, . . . , n (i) d(x)| i(x) i = 1, 2, . . . , n d(x) i(x) (ii) d(x) (iii) (x) F[x] (x)| i(x) i = 1, 2, . . . , n (x)| d(x) i(x) d(x) i(x) F[x] i = 1, 2, . . . , n d(x) = ( 1(x), 2(x), . . . , n(x) ) d(x) = ( 1(x), 2(x), . . . , n(x) ) (x), (x), (x) F[x] d(x) (x), (x), (x) d(x) = = ( ( (x), (x) ), (x) ) d1(x) = ( (x), (x) ) d2(x) = ( ( (x), (x) ) (x) ) = ( d1(x), (x) ) (x) (x) (x) (x) (x)| d1(x) (x) (x)| d2(x) d2(x)| d1(x) d1(x) (x) (x) d2(x) (x) (x) d2(x)| (x) d2(x) (x), (x) (x) (x) d2(x) = ( (x), (x), (x) ) (x), (x), (x) (x), (x) F[x] ( (x), (x) ) = 1 (x), (x), (x) F[x] ( (x), (x) ) = 1 (x)| (x) (x) (x)| (x) ( (x), (x) ) = 1 (x) (x) ( (x), (x) ) = (x)(x) + (x)(x) = 1 (x) (x)(x)(x) + (x)(x)(x) = (x) (x) (x)(x)(x) (x)(x)(x) (x)(x)(x) + (x)(x)(x) = (x) (x), (x), (x) F[x] ( (x), (x) ) = 1 (x)| (x) (x)| (x) (x) (x)| (x)
  18. 18. p(x), p1(x), . . . , pn(x) F[x] F p(x) p1(x) pn(x) c F p(x) = c pi(x) i p(x) p(x)| p1(x) ( p(x), p1(x) ) = 1 ) p(x)| p1(x) ( p(x), p1(x) ) = 1 p(x) p2(x) pn(x) p(x)| p2(x) ( p(x), p2(x) ) = 1 1 i n p(x)| pi(x) p(x) pi(x) c F p(x) = c pi(x) (x) deg( (x) ) = 1 (x) (x) (x) (x) 1(x) 2(x) (x) = 1(x) 2(x) 1(x) 2(x) (x) (x) (x) = c p1(x) p2(x) pn(x) c F[x] pi(x) (x) = c1p1(x)p2(x) pn(x) = c2q1(x)q2(x) qm(x) c1, c2 F p1(x), p2(x), . . . , pn(x), q1(x), q2(x), . . . , qm(x) F qm(x) c1 p1(x) p2(x) pn(x) c F qm(x) = c pi(x) i qm(x) pi(x) qm(x) = pi(x) qm(x) = pn(x) c1 p1(x) p2(x) pn(x) = c2 q1(x) q2(x) qm(x) c1 p1(x) p2(x) pn1(x) = c2 q1(x) q2(x) qm1(x) c1 p1(x) p2(x) pn1(x) (x) c1 = c2 n 1 = m 1 pi(x) = qi(x)
  19. 19. pi(x) (x) = c1p1 1 (x)p2 2 (x) pm m (x) pi(x) i (x) F x4 x2 2 R[x] x4 x2 2 = (x2 2) (x2 + 1) C[x] x4 x2 2 = (x + 2) (x 2) (x i) (x + i) (x), (x) F[x] (x) = c1 p1 1 (x) p2 2 (x) pm m (x) (x) = c2 p1 1 (x) p2 2 (x) pm m (x) i i i = min( i, i ) (x), (x) = p1 1 (x) p2 2 (x) pm m (x) ) (x), (x) F[x] (x) (x)
  20. 20. (x), (x) F[x] (x) = c1 p1 1 (x) p2 2 (x) pm m (x) (x) = c2 p1 1 (x) p2 2 (x) pm m (x) i i Mi = max( i, i ) (x), (x) = pM1 1 (x) pM2 2 (x) pMm m (x) (x) (x) (x) (x) m(x) = ( (x), (x) ) m(x) = [ (x), (x) ] (x), (x) F[x] m(x) F[x] (x) (x) (i) (x)| m(x) (x)| m(x) m(x) (x) (x) (ii) m(x) (iii) (x) F[x] (x)| (x) (x)| (x) m(x)| (x) (x) (x) m(x) (x), (x) F[x] (x) (x) V = { (x) F[x] | (x) (x) (x) } V V m(x) V m(x) V (x) V (x), (x) F[x] (x) = (x) m(x) + (x) deg( (x) ) < deg( m(x) ) (x) deg( (x) ) < deg( m(x) ) (x) (x) (x) (x) m(x) = (x) (x) (x) (x) V m(x) (x) = 0
  21. 21. (x), (x) F[x] (x) (x) c r ( (x), (x) ) = ( c1(x), r1(x) ) (x) (x) ( (x), (x) ) ( (x), (x) ) = (x) (x) (x), (x), (x) F[x] ( (x), (x), (x) ) = ( ( (x), (x) ), (x) ) m1(x) = ( (x), (x) ) m2(x) = ( ( (x), (x) ), (x) ) = ( m1(x), (x) ) m(x) (x) (x) m1(x)| m(x) (x) m2(x)| m(x) m1(x)| m2(x) m1(x) (x) (x) m2(x) (x) (x) (x)| m2(x) m2(x) (x), (x) (x) m(x) m2(x) = ( (x), (x), (x) ) (x) F[x] (x) F[x] 0 = c F c (x) (x) (x) = anxn+an1xn1+ a1x+a0 c (x) (x) (x) = c (x) c ai, i = 0, 1, . . . , n 1(x) = anxn + an1xn1 + a1x + a0, 2(x) = bmxm + bm1xm1 + b1x + b0 F[x] i ai = bi = 0 1(x)| 2(x) 2(x)| 1(x) 1(x) = 2(x)
  22. 22. i(x) F[x] i = 1, 2, . . . , n i j i(x) j(x) (1(x), 2(x), . . . , j(x) , . . . , n(x) ) = ( 1(x), 2(x) , . . . , j1(x), j+1(x) , . . . , n(x) ) i(x) F[x] i = 1, 2, . . . , n (x), (x) F[x] (x) F[x] ( (x), (x) ) = ( (x) + (x) (x), (x) ) (x), p(x) F[x] p(x) F p(x)| (x) ( (x), p(x) ) = 1 p(x)| (x) ( (x), p(x) ) (x), (x) R[x] (x) = 0 1(x), 1(x) R[x] (x) = (x) 1(x) + 1(x) 1(x) = 0 deg( 1(x) ) < deg( (x) ) (x), (x) C[x] (x) = 0 2(x), 2(x) C[x] (x) = (x)2(x)+2(x) 2(x) = 0 deg( 2(x) ) < deg( (x) ) 1(x) = 2(x) 1(x) = 2(x) (x) = x5 x4 x2 + x, (x) = x2 + x 6 R[x] (x) (x) ( (x), (x) ) = (x) (x)+ (x) (x) (x) R[x] 4 ( (x), x2 + 1 ) = 1 ( (x), x2 3x + 2 ) 1 (x), (x) F[x] d(x) = ( (x), (x) ) (x) d(x) (x) d(x) (x), (x) F[x] ( (x), (x) ) (x) (x)
  23. 23. (x), (x), (x) F[x] (x) ( (x) (x), (x) (x) ) = (x) ( (x), (x) ) ( (x) (x), (x) (x) ) = (x) ( (x), (x) ) (x), (x), (x) F[x] ( (x), (x), (x) ) = ( ( (x), (x) ), (x) ) = ( (x), ( (x) (x) ) ) ( (x), (x), (x) ) = ( ( (x), (x) ), (x) ) = ( (x), ( (x) (x) ) ) (x), (x) F[x] ( (x), (x) ) ( (x), (x) ) = (x) (x) (x) (x) (x) = c1 p1 1 (x) p2 2 (x) pm m (x) (x) = c2 p1 1 (x) p2 2 (x) pm m (x) i i i = min( i, i ) ((x), (x)) = p1 1 (x) p2 2 (x) pm m (x) (x) = x5 x4 x2+x, (x) = x2 +x6 R[x]
  24. 24. (x) F[x] (x) = 0 (x) = anxn + an1xn1 + + a1x + a0 F[x] a F anan + an1an1 + + a1a + a0 F (a) a a x a 3x3 2x2 +3x2 3(1/2)3 2(1/2)2 + 3 1/2 2 = 3 1/8 2 1/4 + 3 1/2 2 = 5/8 (x), (x) F[x] a F (a) + (a) = ( + )(a) (a) (a) = ( )(a) F (x) = anxn + an1xn1 + + a1x + a0 F[x] () = 0 2 R x2 x 2 R[x] 22 2 2 = 0 (x) = x2 + 2 R[x] R () = 0 F () = 0 (x) F (x) F[x] a F (x) x a (x) (x), (x) F[x] (x) = (x) (x a) + (x) (x) xa a (x) x a (a) = (a) (a a) + (x) 0 = (a) = (a) (a a) + (x) (x) = 0 a F (x) x a (x) (x) x a (a) a
  25. 25. (x) F[x] (x) = c p1 1 (x) p2 2 (x) pm m (x) a F x a (x) xa pi(x) pi(x) = x a i j F (x) x j j x j j (x) F[x] (x) deg( (x) ) F (x) F[x] 1 F F[x] F (x2 + 1) (x2 + 2) R[x] R R 2 3 (x) F[x] 2 3 (x) F F (x) F 1(x), 2(x) F[x] (x) = 1(x) 2(x) deg( 1(x) ), deg( 2(x) ) deg( (x) ) (x) 2 3 1(x) 2(x) 1 1(x), 2(x) ax + b a, b F a = 0 a1b F (x) x2 + 2 R R C x2 + 2 = (x + 2i) (x 2i) 1 = 2i 2 = 2i R C
  26. 26. z = a + bi z = a bi z1 + z2 = z1 + z2 z1 z2 = z1 z2 z z R (x) = anxn+ an1xn1 + + a1x + a0 (x) (x) = anxn +an1xn1 + +a1x+a0 n z1, z2, . . . , zn C (x) = anxn +an1xn1 + +a1x+ a0 = an(x z1) (x zn) C (x) = anxn + an1xn1 + + a1x + a0 R[x] z z (x) z (z) = anzn + an1zn1 + + a1z + a0 = 0 (z) anzn + an1zn1 + + a1z + a0 = anzn + an1zn1 + + a1z + a0 = 0 an zn + an1zn1 + + a1z + a0 = (z) = 0 (x) = anxn +an1xn1 + +a1x+ a0 R[x] R ax2 + bx + c b2 4ac < 0 Gauss anxn +an1xn1 + +a1x+a0 = 0 B. Fine G. Rosenberger Springer-Verlag 1997
  27. 27. ax + b a = 0 b/a 2k + 1 (x) 2(k + 1) + 1 z z x z x z (x) (xz) (x z) (x) C (x z) (x z) (x) ) 2k + 1 (x) (x) ax2+ bx + c b2 4ac < 0 R (x) deg( (x) ) 3 (x) (xz) (x z) (x) = anxn + an1xn1 + + a1x + a0 F[x] i) (x) 0 a0 ii) (x) 1 n i=0 ai 2x3 3x2 + 6x 5 (x) = anxn + an1xn1 + + a1x + a0 F[x] r F fr : F[x] F fr( (x) ) = (r) F[x] F fr fr f4, r F4[x] f4, r
  28. 28. (x) = anxn + an1xn1 + + a1x + a0 F[x] (x), (x) F[x] n k m = max(n, k) (x) = (x) m+1 a F[x] (a) = (a) (x) = x3 + x2 2x 1 R[x] 2 2 (x) (x)
  29. 29. A (x) V dimF V = m F f : V V (x) = anxn + an1xn1 + + a1x + a0 F[x] anfn + an1fn1 + + a1f + a01V : V V (f) (x) = 2x2 + 3x 2 f : R2 R2 f( (x, y) ) = (x + 2y, x y) (f) : R2 R2 (f)(x, y) = (7x + 6y, 3x + y) (x), (x) F[x] f : V V (f) + (f) = ( + )(f) (f) (f) = ( )(f) A Fmm (x) = anxn+an1xn1+ +a1x+a0 F[x] anAn +an1An1 + +a1A+a0Im Fmm (A) (x) = 2x2 + 3x 2 A = 1 2 1 1 , (A) = 2 1 2 1 1 2 + 3 1 2 1 1 2 1 0 0 1 = 7 6 3 1 . (A) x A (x), (x) F[x] A Fmm (A) + (A) = ( + )(A) (A) (A) = ( )(A) V dimF V = m f : V V u V A = (f : u) m m A u V f : V V A = (f : u) A Fmm (x) = anxn + an1xn1 + + a1x + a0 F[x] u V (A) V dimF V = m f : V V u V (f)
  30. 30. (A) V = R2 u = { (1, 0), (0, 1) } A = 1 2 1 1 f : R2 R2 f( (x, y) ) = 1 2 1 1 x y = (x + 2y, x y) (A) = 7 6 3 1 g : R2 R2 g( (x, y) ) = 7 6 3 1 x y = (7x+6y, 3x+y) (f) ((f) : u) = 7 6 3 1 (f) = g f, g : V V u V A = (f : u) B = (g : u) , F f +g u A+B f g u A B V dimF V = m F f : V V u V A = (f : u) (x) = anxn +an1xn1 + +a1x+a0 F[x] (f) (A) (A) = ((f) : u) V dimF V = m f : V V u, v V A = (f : u) B = (f : v) A B m m P B = P1AP (x) F[x] (A) = ((f) : u) (B) = ((f) : v) (f) (B) = P1(A)P
  31. 31. A B (x) F[x] (A) (B) A = 0 1 1 0 B = 0 1 1 1 (A) (B) (x) = x6 1 A Fnn A = B 0 0 C B Fkk C F(nk)(nk) n An = Bn 0 0 Cn A = B 0 0 C (x) F[x] (A) (A) = (B) 0 0 (C) A Fnn A = B D 0 C B Fkk C F(nk)(nk) D Fk(nk) (x) F[x] (A) (A) = (B) E 0 (C) E Fk(nk) f : R3 R3 f((x, y, w)) = = (x+y, y+z, x+z) (x) = x32x2+3x2 (f) u = { (1, 1, 0), (0, 1, 1), (1, 0, 1) } i) F (x) F[x] A m m (A) (x) ii) i F, i = 1, 2, . . . , m (x) F[x] A m m i (A) i (x)
  32. 32. A Fmm (x) F[x] m2 (A) A Fmm Am2 , Am21, . . . , A, I V : V V 2 = x(x1) (x) F[x] () = 0 (x) = anxn +an1xn1 + +a1x+a0 F[x] () A Fnn k Ak = 0 (x) = (1)k1xk1 + + x2 x + 1 (A) xk + 1 f : V V f (x) = anxn +an1xn1 + +a1x+ a0 F[x] a0 = 0 (f)
  33. 33. V F f : V V = (1, . . . , ) V (f : , ) f f (f : , ) f (f : , ) (f : , ) 1 0 0 0 2 0 0 0 , (f : , ) f Imf (f : , ) (f : , ) 1, . . . , F i = 1, . . . , f(i) = ii. (1)
  34. 34. Imf i i = 0 ker f i i = 0 f f : V V V (f : , ) f V (f : , ) (f : , ) V F f : V V F f v V f(v) = v. v f f : R2 R2 f(x, y) = (x + 2y, 3x + 2y) R v = (x, y) R3 v = (0, 0) f(v) = v (x + 2y, 3x + 2y) = (x, y) (1 )x + 2y = 0 3x + (2 )y = 0. (2) x, y
  35. 35. det 1 2 3 2 = 2 3 4 = ( + 1)( 4) = 0. f = 1 = 4 = 1 2x + 2y = 0 3x + 3y = 0. {(x, x) | x R} = 1 (x, x) x R {0} = 4 3x + 2y = 0 3x 2y = 0. x, 3 2x | x R = 4 x, 3 2 x x R {0} f : R2 R2 f(x, y) = (y, x) R v = (x, y) R2 f(v) = v (y, x) = (x, y) x + y = 0 x + y = 0. R det 1 1 = 2 + 1 = 0 (x, y) = (0, 0) f V
  36. 36. C(R, R) g : R R C(R, R) R g C(R, R) g d : C (R, R) C (R, R), g g g g R ex ex d(ex ) = ex , R d ex f : R2 R2 f(x, y) = (y, x) f 90 W f W R2 dim W = 1 f(W) W v f f(v) = v ( R) W =< v > dim W = 1 f(W) W V F f : V V F f v V f(v) = v (f 1V )(v) = 0 1V : V V f 1V : V V ker(f 1V ) = {0} F ker(f 1V ) = {0} v ker(f 1V ) (f 1V )(v) = 0 f(v) = v f V F f : V V F (i) f (ii) ker(f 1V ) = {0}
  37. 37. f ker(f 1V ) V f V () f V () {0} f : V W f : V V f : V V V F f : V V = (1, . . . , ) V i = 1, . . . , xji F j = 1, . . . , f(i) = x1i1 + x2i2 + + xi. A F i x1i x2i xi . A f (f : , ) (f : ) f A A f V (f : , ) (f : , ) P (f : , ) = P1 (f : , )P. (3) P (1V : , ) (1V : , ) i (1V : , ) y1i y2i yi
  38. 38. i = y1i1 + y2i2 + + yi f : R2 R2f(x, y) = (3x + y, 2x + 4) R2 = (1 = (1, 0), 2 = (0, 1)) = (1 = (1, 1), 2 = (1, 1)). f(1) = (3, 2) = 31 + 22 f(2) = (1, 4) = 11 + 42 (f; , ) = 3 1 2 4 . f(1) = (4, 6) = 51 + (1)2 f(2) = (2, 2) = 01 + 22 (f; , ) = 5 0 1 2 1 = 1 + 2 2 = 1 2 (1V : , ) = 1 1 1 1 . 5 0 1 2 = 1 1 1 1 1 3 1 2 4 1 1 1 1 f, g : V V V (f + g : ) = (f : ) + (g : ) (f g : ) = (f : )(g : ). F (f : ) = (f : ).
  39. 39. V F V f : V V (x) F[x] (f) (A) A = (f : ) ((f) : ) = (A) (x) = anxn + + a1x + a0 (f) = anfn + + a1f + a01V ((f) : ) = (anfn : ) + + (a1f : ) + (a01V : ) = an(fn : ) + + a1(f : ) + a0(1V : ) = an(f : ) + + a1(f : ) + a0(1V : ) = anAn + + a1A + a0I = (A), I = dim V f : R2 R2 R2 A = 3 1 2 4 2f2 3f + 1V 2A2 3A + I2 = 14 10 22 25 I I V F V f : V V A = (f : ) F i) f ii) det(A I) = 0 f ker(f 1V ) = {0}. f 1V : V V ker(f 1V ) = {0} f 1V .
  40. 40. f 1V (f 1V : ) . (f 1V : ) = A I f det(A I) = {0} f det(AI) = 0 A f V f : R2 R2 f(x, y) = (x+2y, 3x+2y) f ((1, 0), (0, 1)) R2 A = 1 2 3 2 det(AI) = det 1 2 3 2 = 2 3 4 f 2 34 = 0 = 1 = 4 f : R3 R3 f(x, y, z) = (4x, 2y 5z, y 2z) f ((1, 0, 0), (0, 1, 0), (0, 0, 1)) R3 A = 4 0 0 0 2 5 0 1 2 R33 . det(A I) = det 4 0 0 0 2 5 0 1 2 = (4 )((2 )(2 + ) + 5) = (4 )(2 + 1). f (4 )(2 + 1) = 0 = 4
  41. 41. f : C3 C3 f(x, y, z) = (4x, 2y 5z, y 2z) F = C f ((1, 0, 0), (0, 1, 0), (0, 0, 1)) C3 A = 4 0 0 0 2 5 0 1 2 C33 . det(AI) = (4)(2+1) f (4 )(2 + 1) = 0 = 4 = i = i R2[x] f : R2[x] R2[x] f((x)) = (x) + (x + 1) (x) = (1, x, x2) R2[x] f A = 1 1 0 0 2 2 0 0 3 . det(A I) = det 1 1 0 0 2 2 0 0 3 = (1 )(2 )(3 ), f 1, 2, 3 A F A : F1 F1 , A(X) = AX F A X F1 AX = X. A F F X F1 X = 0 AX = X A X A
  42. 42. F A F i) A ii) X F1 X = 0 (A I)X = 0 iii) det(A I) = 0 i)ii) AX = X (A I)X = 0 ii)iii) X = 0 (A I)X = 0 det(A I) = 0 A = 1 1 2 1 i) R22 ii) C22 i) R det(A I) = 0 det(A I) = det 1 1 2 1 = 2 + 1 R22 A ii) C det(A I) = 0 det(A I) = 2 + 1 i i = i (A I)X = 0 X = x y (1 i)x y = 0 2x (1 + i)y + 0. (1 i)x y = 0 (x, (1 i)x) x C x (1 i)x x C {0}
  43. 43. = i (A I)X = 0 (i + 1)x y = 0 2x + (i 1)y = 0 (i + 1)x y = 0. x (i + 1)x x C {0} A R33 A = 2 1 0 0 1 1 0 2 4 . det(A I) = det 2 1 0 0 1 1 0 2 4 = (2 )((1 )(4 ) + 2) = (2 )2 (3 ). A = 2 = 3 (AI)X = 0 = 2 (A I)X = 0 X = x y z y = 0 y z = 0 2y + 2z = 0 y = 0 y + z = 0.
  44. 44. (x, 0, 0) x R = 2 x 0 0 x R {0} = 3 (A I)X = 0 x + y = 0 2y z = 0 2y + z = 0 x y = 0 2y + z = 0. (x, x, 2x) x R = 3 x x 2x x R {0} A F (x) F[x] F A X F1 A () (A) X (A) () A X F1 X = 0 AX = X A2X = A(AX) = A(X) = AX = 2X AmX = mX m = 1, 2, . . . (x) = anxn + + a1x + a0 (A)X = (anAn + + a1A + a0I)X = anAn X + + a1AX + a0IX = ann X + + a1X + a0X = (ann + + a1 + a0)X = ()X () (A) X
  45. 45. A = (ij(x)) B = (ij(x)) ij(x), ij(x) F[x] A B F A + B A B AB A B (x)(A) (x) F[x] A A + B = (ij(x) + ij(x)) AB = (ij(x)) ij(x) = k ik(x)kj(x) (x)A = ((x)ij(x)) A + B AB (x)A (x) F[x] A A B F[x] F A F A = (ij(x)) F[x] A = 1 det A = 11(x) > 1 Aij ( 1) ( 1) A i j det A = 11(x) det A11 21(x) det A21 + + (1)+11 det A1 A = x2 0 2 1 x x + 3 x 1 0 x2 2 detA = = x2 det x x + 3 0 x2 2 det 0 2 0 x2 2 + (x 1) det 0 2 x x + 3 = x2 (x(x2 2) 0) 0 + (x 1)(0 2x). F[x]
  46. 46. A = (ij(x)) ij(x) F[x] i = 1, . . . , det A = j=1 (1)i+j ij(x) det Aij i j = 1, . . . , det A = i=1 (1)i+j ij(x) det Aij j i(x) = det A j=1 (1)i+j ij(x) det Aij F[x], det(ij(x)) j=1 (1)i+jij(x) det( (ij) s,t (x)) (ij) s,t (x) = s,t(x) s = i t = j i F a F i(a) = 0 i(x) = 0 det A = det At At A A, B F[x] det(AB) = (det A)(det B) (x) F[x] (x) = det(AB) (det A)(det B) (a) = 0 a F (x) = 0 B A det B = c det A c F[x] c = 0 det A = det B A F A det(A I) = 0 A = (aij) x
  47. 47. A xI = a11 x a12 a1 a21 a22 x a2 a1 a2 a x . det(A xI) x F F A det(A xI) A F det(A xI) A A(x) A R22 A = 2 3 1 1 A(x) = det(A xI) = det 2 x 3 1 1 x = x2 3x 1. B R33 B = 1 3 0 2 2 1 4 0 2 B(x) = det(B xI) = det 1 x 3 0 2 2 x 1 4 0 2 x = x3 + x2 2x 28. A R R A C
  48. 48. B = (ij(x)) F[x] det B = 1(1)(x) ()(x) 1, 2, . . . 11(x) (x) +1 j = 1, 2, . . . , Xj = { S | (1) = j} S = X1X2. . .X = 1 > 1 (1)(1) det B = j=1 (1)i+j 1j(x) det B1j = j=1 (1)i+j 1j(x) Xj 2(1)(x)3(3)(x) ()(x) = j=1 Xj 1(1)(x)2(2)(x) ()(x). S = X1 X2 . . . X j=1 Xj 1(1)(x)2(2)(x) ()(x) = S 1(1)(x)2((2) ()(x). det B = S 1(1)(x)2(2) ()(x) 2 2(x) (x) det B11 = X1 2(2)(x) ()(x) +1 () 11(x)22(x) (x) det B = S 1(1)(x) 2((2) ()(x) +1 X1 X2 . . . X Xi Xj = i = j
  49. 49. A F A(x) = det a11 x a12 a1 a21 a22 x a2 a1 a2 a x = (a11 x)(a22 x) (a x) + , 2 a11 x, a22 x, . . . , a x A(x) (1) A F A(x) A(x) F A A = 1 4 2 3 R22 A20045A+3I A(x) = det 1 x 4 2 3 x = (x 5)(x + 1) A 1 = 5 2 = 1 (x) = x2004 5x + 3 R[x] (5) (1) A2004 5A + 3I (5) = 52004 5 5 + 3 = 52004 22 (1) = (1)2004 5(1) + 3 = 9 (5) = (1) A2004 5A + 3I 2 2 (5) (1) A R A A(x) R R A R A = 1 1 1 1 1 1 .