a21?k?2??a31?3,k ??k?2??ka12?1?a22?k?2??ka32?3, 2?因???1??a11?1? ???3??a13?1?a23?k?2??a33?3,kka12a22ka32a13??a23?k?a33???a11?a 故?在基??1,k?2,?3?下的矩阵为：B2??21?k?a?31 ???1??a12??1??2???a11?a12??1?a31?3, ???3??a23??1??2???a13?a23??1?a33?3, 故?在基??1??2,?1,?3?下的矩阵为： 3?因???1??2???a21?a22???1??2???a11?a12?a21?a22??1??a31?a32??3,a21?a22a12a23???? B3??a11?a12?a21?a22a11?a12a13?a23???a31?a32a31a33???1?12???3.在R3中，试求在基??1,?2,?3?下的矩阵为A???10?1?的线性变换?，其中?122??? ?1??1，0，0?，?2??1，1，0?，?3??1,1,1?.解：显然 ??x1,x2,x3??R3,有?x1,x2,x3???x1?x2??1??x2?x3??2?x3?3 所以 ??x1,x2,x3??A?x1?x2,x2?x3,x3?T ??x1?2x2?3x3,?x1?x2?x3,x1?x2???12?4.对第一题中5）的?，求???关于基?E11,E12,E21,E22?的坐标.?35???a2acabbc???2abadbbd?，解：因?在基下的矩阵为A???acc2adcd???bccdbdd2??????12? 有??在此基下的坐标为?-1,2,-3,5???35???12? 所以???关于基?E11,E12,E21,E22?的坐标??35?
??a2?2ac?3ab?5bc???2?ab?2ad?3b?5bdTT? 为：?y1,y2,y3,y4??A?-1,2,-3,5?????ac?2c2?3ad?5cd????bc?2cd?3bd?5d2????5.在R3中，线性变换?在基?1??1，0，，1??2??1，1，0?，?3??0,1,1?下的矩阵?02?1??? 为A??130?.线性变换?在基?1??1,0,1?,?2??0,1,?1?,?3???1,0,1?下的??201????101??? 矩阵为B??011?.求???，??，??，??????在基??1,?2,?3?下的矩阵.?100???解：由题可知?1??1,?2??2??1,?3??3??2, 即有?1??1,?2??1??2,?3??1??2??3, ?由基?1,?2,?3到基?1，?2，?3的过渡矩阵为?111??? ?011??001??? 故?在基?1，?2，?3下的矩阵为?111??101??111??100????????? ?011??011??011???001?.?001??100??001??111????????? 从而???在基??1,?2,?3?下的矩阵?02?1??100??12?1??????? ?130???001???131???201??111???112??????? 从而??在基??1,?2,?3?下的矩阵?02?1??100???1?11??????? ?130??001???103???201??111???111??????? 从而??在基??1,?2,?3?下的矩阵?100??02?1??12?1??????? ?001??130????201??111???201???150??????? 从而??????在基??1,?2,?3?下的矩阵
?1?100???02?1??100???12?1??????????? ?001???130???001?????112??111????201??111???162???????????6.给定线性空间F3的的两个基 ?1??1，01，，，，0?，?3??1,1,1?,?1??1,2,?1?,?2??2,2,?1?,?3??2,?1,?1???2??21 设?是F3的线性变换，且???i???i,i?1,2,3.1?写出由基??1,?2,?3?到??1,?2,?3?的过度矩阵；2?写出?在基??1,?2,?3?下的矩阵；3?写出?在基??1,?2,?3?下的矩阵.?1?121??122?2?????解：1?由题有A??011?,B??22?1?,而A?1??12?101???1?1?1???1?????2 ?有由基??1,?2,?3?到??1,?2,?3?的过度矩阵为3??2?322??33? T?A?1B??122?1?1??522?? 2?由题有???1???1??1,2,?1???2?1??2??331 ???2???2??2,2,?1???32?1?2?2?2?331 ???3???3??2,?1,?1??32?1?2?2?2?33??2?322??33? ?有?在基??1,?2,?3?下的矩阵为C??122?11??122??33311 3?由于?1??2?1??2??3，?2??32?1?2?2?2?3，?3?2?1?2?2?2?3?112??0?12?1?12? 所以有???1?????2?1??2??3???2?1??2??333311 ???2?????32?1?2?2?2?3???2?1?2?2?2?333311 ???3????32?1?2?2?2?3??2?1?2?2?2?3?12?1??? ?有?在基??1,?2,?3?下的矩阵为D???2?13??062????15?115???7.设F上的三维线性空间的线性变换?在基??1,?2,?3?下的矩阵是?20?158?.?8?76??? 1?求?在基?1?2?1?3?2??3,?2?3?1?4?2??3,?3??1?2?2?2?3,下的矩阵. 2?设?=2?1??2??3,求????在基??1,?2,?3?下的坐标。解：由题可得???1??2???1??3???2?????3??2?1?3?2??3??1 ???2??3???1??4???2?????3??6?1?8?2?2?3?2?2 ???3?????1??2???2??2???3??3?1?6?2?6?3?3?3?100??? ??在基??1,?2,?3?下的矩阵为A??020??003????2??15?115??2??????? 由题可得????????1,?2,?3??1????1,?2,?3??20?158??1???1??8?76???1???????
?231??15?115??2???5????????????1,?2,?3??342??20?158??1????1,?2,?3??8??112??8?76???1??0?????????8.设A、B是n阶方阵，且A可逆.证明：AB?BA.证明：因A可逆，则A?1也可逆 故有BA?A?1?AB?A 由定义2有AB?BA?A0??B0?9.设A?B，C?D.证明：?????.?0C??0D?证明：?A?B，C?D ??T1,T2,s..tB?T1?1AT1,D?T2?1AT20??B0??B0??T1?1AT1 ?对?有???????10D0D0TAT?????22??B0??T1?10??A0??T10? 即????????1??0D???0T2??0C??0T2??A0??B0? 故由相似的定义有??????0C??0D?10.证明：数域F上的一个上三角矩阵必与一个下三角矩阵相似.证明：设??L?V?,??1,?2,?,?n?是F上的线性空间V的基?a11a12?a1n???0a?a222n? 设A是?在基??1,?2,?,?n?下的矩阵，且A?????????00?ann?????1??a11?1?????2??a12?1?a22?2? 则???????n??a1n?1?a2n?2???ann?n?????n??ann?n?a?n?1?n?n?1???a1n?1?????n?1??0?n?an?n?1??n?1???a1?n?1??n 而?在基??n,?n?1,?,?1?下有???????1??0?0???a11?1??ann??a?n?1?n 即有???n,?n?1,?,?1????n,?n?1,?,?1?????a1n? ???n,?n?1,?,?1?B0a22?a1?n?1?0???0?????a11???
?1
第6章习题答案
