复习笔记Day4

2022-08-25 08:45 作者:臥煙鈴 0人读过 | 我要投稿

今天给大家带来了两道高代的题

这次为了手机上也能够很好的查看，公式就没有编辑得特别长一串了

8.(中山大学2017)

$%E8%AE%BEV%E6%98%AF%E4%B8%80%E4%B8%AAn%E7%BB%B4%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4%EF%BC%8C%5Csigma%5Cin%20EndV%E3%80%82$

$%E8%8B%A5%5Cexists%20%E5%8D%95%E4%BD%8D%E5%90%91%E9%87%8F%5Ceta%E4%BD%BF%E5%BE%97%5Csigma(%5Calpha)%3D%5Calpha-2(%5Calpha%2C%5Ceta)%5Ceta$

$%E5%88%99%E7%A7%B0%5Csigma%E4%B8%BA%E9%95%9C%E9%9D%A2%E5%8F%8D%E5%B0%84%EF%BC%8C%E8%BF%99%E9%87%8C(%5Ceta%2C%5Calpha)%E8%A1%A8%E7%A4%BA%5Ceta%E4%B8%8E%5Calpha%E7%9A%84%E5%86%85%E7%A7%AF.$

$(1)%E8%8B%A5%5Csigma%E6%98%AF%E9%95%9C%E9%9D%A2%E5%8F%8D%E5%B0%84%EF%BC%8C%E8%AF%81%E6%98%8E%3A$

$V%E6%9C%89%E6%AD%A3%E4%BA%A4%E5%88%86%E8%A7%A3V%3Dker(id_v%2B%5Csigma)%5Coplus%20ker(id_v-%5Csigma)%20.$

$(2)%5Calpha%E4%B8%8E%5Cbeta%E6%98%AFV%E4%B8%8A%E7%BA%BF%E6%80%A7%E6%97%A0%E5%85%B3%E4%B8%A4%E5%8D%95%E4%BD%8D%E5%90%91%E9%87%8F%EF%BC%8C$

$%E6%B1%82%E4%B8%80%E4%B8%AA%E9%95%9C%E9%9D%A2%E5%8F%8D%E5%B0%84%5Ctau%EF%BC%8C%E4%BD%BF%E5%BE%97%5Ctau(%5Calpha)%3D%5Cbeta.$

$(1)%E8%AF%81%E6%98%8E%3A$

$%E8%AE%BE%E6%AD%A4%E6%97%B6%5Cexists%20v%E4%BD%BF%E5%BE%97%5Csigma(%5Calpha)%3D%5Calpha-2(%5Calpha%2Cv)v$

$%E4%B8%8D%E5%A6%A8%E4%BB%A4%5Csigma(%5Calpha)%2Bid_v(%5Calpha)%3D0$

$%E5%BE%97%E5%88%B00%3D2%5Calpha-2(%5Calpha%2Cv)v%5Cimplies%20%5Calpha%3D(%5Calpha%2Cv)v$

$%E4%BA%8E%E6%98%AF%5Calpha%E4%B8%8Ev%E7%BA%BF%E6%80%A7%E7%9B%B8%E5%85%B3%EF%BC%8C%E5%8D%B3ker(%5Csigma%2Bid_v)%3Dspan%5Cleft%5C%7B%5Calpha%20%5Cright%5C%7D$

$%E6%88%91%E4%BB%AC%E4%B8%8D%E5%A6%A8%E8%AE%B0span%5Cleft%5C%7B%5Calpha%20%5Cright%5C%7D%E4%B8%BAW$

$%E4%BB%A4%5Csigma(%5Calpha)-id_v(%5Calpha)%3D0%5Cimplies%200%3D(%5Calpha%2Cv)v$

$%E4%BA%8E%E6%98%AFker(%5Csigma-id_v)%3D%5Cleft%5C%7B%5Calpha%7C%5Cforall%20%5Calpha%5Cin%20V%2C(%5Calpha%2Cv)%3D0%5Cright%5C%7D%3DU$

$%E8%80%8C%5Cforall%20%5Cbeta%5Cin%20W%5C%20%5Cexists%20k%5Cin%20%5Cmathbb%7BR%7D%2Cst%20%5C%20%5Cbeta%3Dkv$

$%E6%95%85%5Cforall%20%5Cbeta%5Cin%20W%5C%20%2C%5Cforall%5Calpha%20%5Cin%20U%E6%9C%89(%5Cbeta%2C%5Calpha)%3Dk(v%2C%5Calpha)%3D0$

$%E8%AE%BE%5Cexists%20%5Calpha%5Cin%20W%2C%E4%B8%94%5Calpha%20%5Cin%20U%2C%E5%88%99(%5Calpha%2C%5Calpha)%3D0%5Cimplies%20%5Calpha%3D0$

$%E6%98%93%E7%9F%A5dimU%2BdimV%3Dn%2C%E6%95%85U%5Coplus%20W%3DV$

$(2)%E8%A7%A3%3A$

$%E8%AE%BE%E5%AD%98%E5%9C%A8%5Ctau%EF%BC%8C%E4%BD%BF%E5%BE%97%5Ctau(%5Calpha)%3D%5Calpha-2(%5Calpha%2Cu)u%3D%5Cbeta$

$%5Cimplies%20%5Calpha-%5Cbeta%3D2(%5Calpha%2Cu)u$

$%E7%94%B1%20%5Calpha-%5Cbeta%3D2(%5Calpha%2Cu)u%E6%88%91%E4%BB%AC%E5%BE%97%E7%9F%A5%20%5Calpha-%5Cbeta%E4%B8%8Eu%E7%BA%BF%E6%80%A7%E7%9B%B8%E5%85%B3$

$%E4%BB%A4u%3D%5Cfrac%7B1%7D%7B%7C%7C%5Calpha-%5Cbeta%7C%7C%7D(%5Calpha-%5Cbeta)%2C%7C%7C%5Calpha-%5Cbeta%7C%7C%3D%5Csqrt%7B(%5Calpha-%5Cbeta%2C%5Calpha-%5Cbeta)%7D%20$

$%5Calpha-2(%5Calpha%2Cu)u%3D%5Calpha-2(%5Calpha%2C%5Cfrac%7B(%5Calpha-%5Cbeta)%7D%7B%7C%7C%5Calpha-%5Cbeta%7C%7C%7D)%5Cfrac%7B(%5Calpha-%5Cbeta)%7D%7B%7C%7C%5Calpha-%5Cbeta%7C%7C%7D$

$%3D%5Calpha-2(%5Calpha%2C%5Calpha-%5Cbeta)%5Cfrac%7B(%5Calpha-%5Cbeta)%7D%7B(%5Calpha-%5Cbeta%2C%5Calpha-%5Cbeta)%7D$

$%3D%5Cfrac%7B(%5Calpha-%5Cbeta%2C%5Calpha-%5Cbeta)%5Calpha-2(%5Calpha%2C%5Calpha-%5Cbeta)(%5Calpha-%5Cbeta)%7D%7B(%5Calpha-%5Cbeta%2C%5Calpha-%5Cbeta)%7D$

$%3D%5Cfrac%7B%5B2-2(%5Calpha%2C%5Cbeta)%5D%5Calpha-2%5B1-(%5Calpha%2C%5Cbeta)%5D(%5Calpha-%5Cbeta)%7D%7B2-2(%5Calpha%2C%5Cbeta)%7D%3D%5Cbeta$

$%E6%95%85%E5%AD%98%E5%9C%A8%5Ctau(v)%3Dv-2(v%2C%5Cfrac%7B(%5Calpha-%5Cbeta)%7D%7B%7C%7C%5Calpha-%5Cbeta%7C%7C%7D)%5Cfrac%7B(%5Calpha-%5Cbeta)%7D%7B%7C%7C%5Calpha-%5Cbeta%7C%7C%7D$

$%E4%BD%BF%E5%BE%97%5Ctau(%5Calpha)%3D%5Cbeta$

$%5Cmathbb%7BQ.E.D%7D$

6.(中山大学2017)

$%E8%AE%BEA%E6%98%AF%E6%95%B0%E5%9F%9FF%E4%B8%8An%E9%98%B6%E6%96%B9%E9%98%B5$

$%E8%AF%81%E6%98%8E%3Arank(A)%3D1%5Ciff%20%5Cexists%20%5Calpha%2C%5Cbeta%5Cneq0%2Cst%20%5C%20A%3D%5Calpha%5Cbeta%5ET$

$%E8%AF%81%3A$

$%E8%8B%A5rankA%3D1%2C%E5%88%99%E5%B0%86A%E8%A1%8C%E5%88%86%E5%9D%97%2C%E8%AE%BEA%3D(%5Calpha_1%2C%5Calpha_2%2C%5Cldots%2C%5Calpha_n)$

$%E5%88%99rank(%5Calpha_1%2C%5Calpha_2%2C%5Cldots%2C%5Calpha_n)%3D1$

$%5Cexists%20%5Calpha%5Cneq0%2Cst%5C%20%5Calpha_i%3Dk_i%5Calpha%2Ci%3D1%2C2%2C%5Cldots%2Cn$

$%E5%88%99A%3D(%5Calpha_1%2C%5Calpha_2%2C%5Cldots%2C%5Calpha_n)%3D(k_1%5Calpha%2Ck_2%5Calpha%2C%5Cldots%2Ck_n%5Calpha)$

$%3D%5Calpha(k_1%2Ck_2%2C%5Cldots%2Ck_n)%5Cimplies%20%5Cbeta%5ET%3D(k_1%2Ck_2%2C%5Cldots%2Ck_n)$

$%E8%8B%A5%5Cbeta%5ET%3D(k_1%2Ck_2%2C%5Cldots%2Ck_n)%3D0%EF%BC%8C%E5%88%99A%3D0%EF%BC%8C%E6%95%85%5Cbeta%5Cneq0$

$%E7%9B%B8%E5%8F%8D%E7%9A%84%2C%5Cforall%5Calpha%2C%5Cbeta%5Cneq0%EF%BC%8C%E8%AE%BE%5Cbeta%5ET%3D(k_1%2Ck_2%2C%5Cldots%2Ck_n)$

$%E5%88%99%5Calpha%5Cbeta%5ET%3D%5Calpha(k_1%2Ck_2%2C%5Cldots%2Ck_n)%3D(k_1%5Calpha%2Ck_2%5Calpha%2C%5Cldots%2Ck_n%5Calpha)%3DA$

$%E8%80%8Crank(k_1%5Calpha%2Ck_2%5Calpha%2C%5Cldots%2Ck_n%5Calpha)%3D1%5CRightarrow%20rankA%3D1$

$%5Cmathbb%7BQ.E.D%7D$

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