6 6.1 6.2 6.3 6.4 6.5 6.6 6.7
Vector SpacesLinear Spaces. . . ..
6.1
A+B=B+A(A+B)+C= A+( B+C) OA=AA+(-A)=Oa(A+B)= aA+Ab(a+b)B=a B +Bb(ab)A=a(b)A8. 1AA
2 RV3.R8.x,y,z . . 8.
1FV.VV(c1) V()u,vV, u+vV.(c2) FV ()Fa Vv, avV.(a1) u+v= v +uuvV.(a2) u+(v+w)= (u+v)+w, uvwV.(a3) Vo, v+o= v Vv.(a4) Vv, Vu u+v= 0. uv.
(m4) 1u= u uV.
4 FF[x]F.
(a2) [f(x)+g(x)]+h(x)= f(x)+ [g(x) +h(x) ], (a3) 0. (a4) f(x)- f(x).
5 C[a,b][a,b]R.3.6 1FF.2RQRC
7 R, R+()
3.4.
51082.
.
1 0.2 v- v.
6.2 1.2.3.
1VFWV . 1WWWV. 2WFaaWW .
2WFV.W V . 3WFV.W V WV . VFV
V{0}VV VVVV
V2V2V3V3
F [x]nF [x] [a,b]C [a,b]
1 A0 = 0
4WWVa,bF,W a+bW
1W1W2VW1W2V.2 {Wi }V. V.3W1W2
W1+W2V,W1W2 .
6.36.3.1 6.3.2 6.3.3 6.3.4 1 23
6.3.1 1
6.3.2 2
1 1=1,2,32=2,4,63=3,5,-42 1=1,-2,32=2,1,03=1,-7,93 F [x] n ,
6.3.1 6.3.2 6.3.3
6.3.4 6.3.5
6.3.3
6.3.6 (6.3.7
6.3.4
5F36.3.8 .
6.4 6.4.1 6.4.26.4.3 6.4.4 12
6.4.1
1
2Fn.
6.4.1
6.4.2 1 VV
3 4
dimnnmnmn
.. 5 x...3 nn
..
6.4.3 .. dimdimdimdim6.4.4 .. 6 VWW . V W W
6.4.7 n VW , WW , dimV = dimW + dimW.
6.5 6.5.1 6.5.2 6.5.3 1..2..
6.5.1 1
1 2 3
.
6.5.2 2
4
6.5.3 6. 5. 2
(4)
5 ,6
6.6 6.6.1 6.6.2 6.6.3 1..2.. .
6.6.1 1 if VW6.6.2
2. f VW
6.6.3 1 2 .3
67 6.7.16.7.2 123 .
6.7.1 Fmn
iPAA iiAQA
4.
6.7.2F1 F1
A2 2
3
3 1
,
.
4 F4 AX = 0 AX = B44