Chapter 1 Vector Spaces

Transcription

1 Chpter Vetor pes - Vetor pes Ler Comtos Vetor spe V V s set over fel F f V F! + V. Eg. R s vetor spe. For R we hek -4=-4-4R -7=-7-7R et. Eg. how tht the set of ll polomls PF wth oeffets from F s vetor spe. Proof g f PF F f+g=f+gpf for V F where where 4 uspe A suset of V s suspe of V Eg. R s suspe of R. For R we hek +=++R -= --R R +--= the --R et. Eg. V {} re oth suspe of V. Theorem A terseto of suspes of vetor spe V s suspe of V. Theorem re suspes of V the s suspe or. um of two sets + + ={+ }.. Eg. Let ={os os os...} ={s s s...} the + ={os+s os+s os5+s...}. Theorem re suspe of V the + s the smllest suspe tht ots oth. Dret um f ={} = + Eg. F = where F F Eg. PF= where 5 4 g f m m

2 Eve futo f-=f O futo f-=-f Eg. -7 os re eve futos ut s4 re o futos. Theorem re the set of ll eve futos the set of ll o futos FCC respetvel. The FCC=. Proof. re oth suspes of FCC. f f=f-=-f f= ={} Let h g g the h FCC=. g g Theorem Let e two suspes of vetor spe V over F the V= V!! suh tht = +. Proof uppose = + = + - = = - ={ } = =. Eg. Let eote the - the - the z-s respetvel. The R = j ={}. R =++ where j. R s uquel represete s ret sum of. Eg. Let eote the - the z-ples respetvel. The R = + ={z =z=} {}. R =+=+ where R ot e uquel represete s ret sum of. Theorem s suspe of V re elemets of the s elemet of for over F. Proof = t hols efto. uppose =k k k k k k k s elemet of the =k+ s lso elemet of efto. the proof s omplete. Ler omto = It s lle the ler omto of V where F s oempt suset of V. p the suspe geerte the elemets of The suspe ossts of ll ler omtos of elemets of. Eg. ={.} the p={+ F}={ }.

3 Theorem s oempt suset of V p s suspe of V. p s the smllest suspe of V otg the sese tht p s suset of ol suspe of V tht ots. Eg. F = M R=p. Eg. =++= + + F R =p{}=p{}. Eg. Plot p p. [6 台科大電子所 ] ol. p = le of = p = the -s. p p uo of = the -s. Note p p p +p = s the -ple. Eg. Let ={ } e lerl epeet set ther oeffets e selete from {}. How m elemets re there p? ol. If p = elemets. Theorem pφ={}. Theorem A suset of vetor spe V s suspe of V p=. Proof p s suspe of V = p s suspe of V. "" "" If p= s suspe of V. = p s the smllest suspe of V otg s ot suspe of V. It s otrtor to the sttemet. p= =. Theorem If re suspe of V the p p. p p p. [ 台大電研 ] Proof = p F p p p p p p p p p p

4 Theorem If re suspe of V the p =p +p. [ 台大電研 ] Proof p p p uppose p =p +p + where s epeet of. Let V V = p = p + p - Ler Depeee Ler Iepeee Ler epeee & ler epeee For = f t. re ll zeros the s lerl epeet; otherwse s lerl epee Eg re lerl epeet euse of +-6-4= ut re lerl epeet euse of ol += Theorem V s vetor spe V. If s lerl epeet the s lso lerl epeet. If s lerl epeet the s lso lerl epeet. Bss A ss β for vetor spe V s lerl epeet suset of V tht geertes V. Dmeso mv The uque umer of elemets eh ss for V. Theorem If V= the mv=m +m. Eg. R we hve = + +. Thus { } s the ss of R mr =. ss of m. ol. Bss of {--} m=. Eg. R we hve = +. Thus { } s the ss of R mr =. Eg. For ={ 4 5 R = = 4 } f =. et =r =s 5 =-r-s set = 4 =t. 4 5 =rtst-r-s=r-+t+s-

5 Eg. Let V=p{A A A A 4 } where A = A = A = = ].. A4. F ss for V. [5 台大電研 ol. A +A +A +A 4 = = r r r A A A A 4 re lerl epeet. I ft.5a +.5A =A e rop A A + A + A 4 = = A A A 4 re lerl epeet {A A A 4 } s the ss of V. Theorem β={ } s ss for V V e uquel epresse s ler omto of vetors β. Proof If = = β s lerl epeet Theorem s lerl epeet suset of V let V ut. The {} s lerl epeet p. Eg. how tht se β={ } e ss R the β ={ } s lso ss R. [ 文化電機轉學考 ] Proof et. If the re lerl epeet re lerl epeet. mr = β s ss of R. Aother metho = β s lso ss of R et

6 Eg. Determe whether the gve set of vetors s lerl epeet? [ 交大電信所 ] {} R. {--5-6} R. {--54} R. ol. et = Lerl epeet. 4 vetors R Lerl epeet. et Lerl epeet. Eg. Are lerl epeet? [99 中央土木所 ] ol.. If <. If < hol = = Lerl epeet. 5 Eg. Gve mtr A= set of mtres ={ }. Determe f s lerl epeet suset of M the vetor spe of ll mtres? Represet the mtr A s ler omto of the vetors the set. ht re the orrespog oeffets? [ 台大電研 ] ol. Let = ==== ={ } s lerl epeet suset of M. e 5 e 5 5 Let = e +f +g +h e g h f e f h g f g h Eg. re fte-mesol suspe of V m =m m = m the m m + m+. [ 台大電研 ] Proof. m m =. m + =m +m -m m m + m+

7 Eg. Let v e the sp of the set of vetors ={-5}. ht s the meso of ν? C we use s ss of v? [6 台科大電研 ] ol. Let -++5= =- =- = ={-5} s lerl epeet. ={5} s lerl epeet mv=. No! Trspose of m mtr M M t A m mtr M t whh M t j =M j. Eg. M= 5 4 the M t 5 = I Mtl lguge we use the followg strutos to ot the trspose of mtr >>A=[5;] A = 5 >>C=A C = 5 mmetr mtr M=M t ; tht s M j =M j. kew smmetr mtr M=-M t ; tht s M j =-M j j M j = =j..6 Eg. A=.5 7 s smmetr mtr. B= s skew smmetr mtr Eg. how tht the set of ll squre mtres e eompose to the ret sum of the set of the smmetr mtres tht of the skew-smmetr oes. [ 文化電機轉學考 ] Proof. The set of the smmetr mtres the set of the skew-smmetr mtres re oth suspes of M F. A A=A t =-A t A = ={} Let B C A A t A A t the B C M F=.

8 Eg. The set of smmetr mtres M F s suspe. F ss for m. [ 文化電機轉學考 ] ol. where j= j. 4. m Note The meso of set of skew-smmetr mtres M F s. Eg. ht re the mesos of the set of ll the 5 5 smmetr mtres tht of ll the 5 5 skew-smmetr oes respetvel? ol. Dmesos of the set of ll the 5 5 smmetr mtres= 5 5 =5 Dmesos of the set of ll the 5 5 skew-smmetr mtres= 5 5 =