天津大學(xué)工程數(shù)學(xué)基礎(chǔ)新版習(xí)題答案

1、1工程數(shù)學(xué)基礎(chǔ)習(xí)題解答3)1212(kkx????從而從而???????1)1212(nnnx，故????????1]22[)1212(nnn.因此因此????????1)1212(]22[nnn.3.supsupsup.AAA?????????證設(shè)且要證唯一只需證明即可supsup.inf.AAAAA????????????????????因?yàn)槭亲钚∩辖缍堑纳辖绻视忠驗(yàn)槭亲钚∩辖缍堑纳辖绻室虼祟愃频乜梢宰C明是唯一的4.證設(shè)??DY

2、???是線性空是線性空間X的一族子空間的一族子空間要證要證DYX????也是的線性子空間.顯然顯然DY??????，z只需證明只需證明.DYX????對(duì)的線性運(yùn)算是封閉的事實(shí)上，事實(shí)上，DxyY??????及???從而對(duì)每一個(gè)從而對(duì)每一個(gè)D??有xyY??故xyY???xY???.于是于是DxyY??????DxY??????.因此因此DY????是X的線性子空間的線性子空間.5.WfgW?????證顯然包含零多項(xiàng)式故非空；又及，有()

3、(0)()(0)(0)(0)(0)(0)[(0)(0)][(0)(0)]000fgfgfgfgffgg???????????????????即()(0)()(0)(0)(0)[(0)(0)]00.fgWfffffffW????????????????????即[01].nWP所以是的線性子空間1111021121001121[01]()()2.(0)(0)00()(1).nnnnnnnnnnnfWPfxaxaxaxafxnaxaxaff

4、aaaafxaxaxaxax???????????????????????????????設(shè)則由得即故23(1)dim.nxxxxWWn??由上可知是的一個(gè)基故6.1(1)(0)0.()00.TTTxTTx?????“”:因?yàn)槭蔷€性的故有于是若則由存在知是單射從而有1TT??“”:要證存在只需證明是單射：121212121212(()()()()00.xxXTxTxTxxTxTxxxxxT??????????當(dāng))即時(shí)由條件得即故是單射1

5、112121211221122(2)s.t.()().yyYxxXyTxyTxxTyxTy??????????????及即于是有1111111221122112211221122()[()()][()]()()TyyTTxTxTTxxxxTyTy???????????????????????1:.TYX??故是線性的7.2222:.BA????解首先驗(yàn)證是線性的然后求其在即下的矩陣221212XXkk??????由的定義，有100100