[研究生入學(xué)考試]線性代數(shù)-資料下載頁(yè)
2025-01-19 15:39本頁(yè)面
【正文】 ?????aaaaaaAba11001110111,12則若?????????????11001110111aa??????????????110020221011aa??????????????110020221001a???????????121321xaxx此時(shí)有唯一解.33293132321取不同值時(shí)解的情況在討論線性方程組例aaxxaxxxxx????????????? ??????????????aaAb3011103121:解???????????????32201103121aa??????????????100011023301aaa當(dāng) a≠1時(shí) ,方程組無(wú)解 a=1時(shí) ? ???????????????100011023301aaaAb?????????? ??000011101301??????????cxc1xc31x321方程組有無(wú)窮多解 ????????????????332222132143214321xxxxxxxxxxx? ????????????????303222121111111: Ab解?????????? ???121001210011111前半部分已無(wú)法化為單位塊 例 10 解線性方程組 若改為方程組為 ????????????????323222123142314231xxxxxxxxxxx則增廣矩陣 ??????????????302322112111111?????????? ???120221202211111若允許交換兩列 ,可寫(xiě)為 ? ??????????? ???121001210011111Ab?????????? ???120221202211111?????????? ???000001202203101?????????????241223211213cxcxcxccx方程組有無(wú)窮多解?????????????242312211213cxcxcxccx或?qū)憺閤3 x2 也可換第二 ,第四列 ? ??????????? ???121001210011111Ab?????????? ???102010201111?????????? ??0000010120101 2323?????????? ??0000001010121212323??????????????22131212142123231cxcxcxccx無(wú)窮多解為x4 x2 ? ? ???????? ??31201211.Ab例如解的不同寫(xiě)法交換未知數(shù)次序可得到???????? ??2321101211???????? ??232125231001???????????????cxcxcx32121232325解為? ? ???????? ??31201211Ab若寫(xiě)為???????? ??32101121x3 x2 ???????? ???32107301???????????cxcxcx2312337解為???????????????????0. ... ..0. ..0. ..3221122221211212111nmnmmnnnnxaxaxaxaxaxaxaxaxa程組常數(shù)項(xiàng)全為零的線性方定義稱為齊次線性方程組 . 寫(xiě)成矩陣形式為 Ax=0 )0. ..( 21 ???? nxxx齊次方程組一定有解定理 2 齊次線性方程組 A mxn x=0有非零解 (無(wú)窮多解 ) 的充要條件是 r( A)n 由定理 1 可得 故為方陣時(shí)因?yàn)?,)(0, nArAA n x n ???推論 1 齊次線性方程組 A nxn x=0有非零解 的充要條件為 |A|=0 這是第一章用過(guò)的結(jié)論 . 又由于 mn時(shí) ,r( A mxn)≤ mn,故得 推論 2 齊次線性方程組中未知數(shù)個(gè)數(shù)多于 方程個(gè)數(shù)時(shí) ,一定有非零解 (無(wú)窮多解 ). .02 02321321 必有非零解如?????????xxxxxx?????????????????0220220411432143214321xxxxxxxxxxxx解齊次線性方程組例解 :由推論知它必有無(wú)窮多解 ? ?????????????????0211201212041110A??????????????0103300701004111?????????????0113000701003101???????????01000701003101311?????????????0113000701003101???????????01000701003101311?????????? ??010007010000131132???????????????cxcxcxcx4321311732無(wú)窮解為.?002021232121321并求出所有解為何值時(shí)有非零解在齊次線性方程組例axxxaxxxxax?????????????解 :系數(shù)行列式為 1110212?? aaA11102031??? aa62 ??? aa? ?? ?23 ??? aa故當(dāng) a≠3且 a≠2時(shí) ,方程組僅有零解 當(dāng) a=3或 a=2時(shí) ,方程組有非零解 . a=3時(shí) ? ?????????????0111003201230A?????????? ??012300320111??????????????025002500111????????????00000250001 53????????????00000100015253?????????????cxcxcx3215253所有解為a=2時(shí) ? ???????????????0111002201220A??????????????012200220111?????????????030002000111?????????? ??000001000011?????????? ??000000100101x3 x2 ????????0321xcxcx所有解為
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