线性代数：求一个极大线性无关组，并将其余向量用此极大线性无关线性组表示 求下列向量组的一个极大无关组，并将其余向量用此极大无关组线性...
\u7ebf\u6027\u4ee3\u6570\uff1a\u6c42\u4e0b\u5217\u5411\u91cf\u7ec4\u7684\u79e9\u548c\u4e00\u4e2a\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\uff0c\u5e76\u5c06\u5176\u4f59\u5411\u91cf\u7528\u6b64\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u7ebf\u6027\u8868\u793a\uff1f\u03b11A = (a1,a2,a3,a4) =
[1 2 -1 3]
[0 1 0 1]
[1 1 0 1]
[0 2 0 2]
\u884c\u521d\u7b49\u53d8\u6362\u4e3a
[1 1 0 1]
[0 1 -1 2]
[0 1 0 1]
[0 2 0 2]
\u884c\u521d\u7b49\u53d8\u6362\u4e3a
[1 1 0 1]
[0 1 -1 2]
[0 0 1 -1]
[0 0 2 -2]
\u884c\u521d\u7b49\u53d8\u6362\u4e3a
[1 1 0 1]
[0 1 -1 2]
[0 0 1 -1]
[0 0 0 0]
\u5219\u5411\u91cf\u7ec4\u7684\u79e9\u4e3a3,a1,a2,a3 \u4e3a\u4e00\u4e2a\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4.
\u518d\u884c\u521d\u7b49\u53d8\u6362\u4e3a
[1 1 0 1]
[0 1 0 1]
[0 0 1 -1]
[0 0 0 0]
\u884c\u521d\u7b49\u53d8\u6362\u4e3a
[1 0 0 0]
[0 1 0 1]
[0 0 1 -1]
[0 0 0 0]
\u5f97 a4=a2-a3
\u91cd\u8981\u5b9a\u7406
\u6bcf\u4e00\u4e2a\u7ebf\u6027\u7a7a\u95f4\u90fd\u6709\u4e00\u4e2a\u57fa\u3002
\u5bf9\u4e00\u4e2a n \u884c n \u5217\u7684\u975e\u96f6\u77e9\u9635 A\uff0c\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u77e9\u9635 B \u4f7f AB = BA =E\uff08E\u662f\u5355\u4f4d\u77e9\u9635\uff09\uff0c\u5219 A \u4e3a\u975e\u5947\u5f02\u77e9\u9635\uff08\u6216\u79f0\u53ef\u9006\u77e9\u9635\uff09\uff0cB\u4e3aA\u7684\u9006\u9635\u3002
\u77e9\u9635\u975e\u5947\u5f02\uff08\u53ef\u9006\uff09\u5f53\u4e14\u4ec5\u5f53\u5b83\u7684\u884c\u5217\u5f0f\u4e0d\u4e3a\u96f6\u3002
\u77e9\u9635\u975e\u5947\u5f02\u5f53\u4e14\u4ec5\u5f53\u5b83\u4ee3\u8868\u7684\u7ebf\u6027\u53d8\u6362\u662f\u4e2a\u81ea\u540c\u6784\u3002
\u77e9\u9635\u534a\u6b63\u5b9a\u5f53\u4e14\u4ec5\u5f53\u5b83\u7684\u6bcf\u4e2a\u7279\u5f81\u503c\u5927\u4e8e\u6216\u7b49\u4e8e\u96f6\u3002
\u89e3: (a1^T,a2^T,a3^T,a4^T) =
2 -1 3 8
1 1 1 5
1 7 -1 9
1 10 -2 11
r1-2r2,r3-r2,r4-r2
0 -3 1 -2
1 1 1 5
0 6 -2 4
0 9 -3 6
r3+2r1,r4+3r1,r2-r1
0 -3 1 -2
1 4 0 7
0 0 0 0
0 0 0 0
\u6240\u4ee5 a1,a3 \u662f\u4e00\u4e2a\u6781\u5927\u65e0\u5173\u7ec4
a2 = 4a1 - 3a3a4 = 7a1 - 2a3
= 1 -1 5 -1
1 1 -2 3
3 -1 8 1
1 3 -9 7
1 -1 5 -1
0 2 -7 4
0 2 -7 4
0 4 -14 8
1 -1 5 -1
0 2 -7 4
0 0 0 0
0 0 0 0
1 0 3/2 1
0 1 -7/2 2
0 0 0 0
0 0 0 0
{a,b} 极大线性无关组 c=3/2a-7/2b d=a+2b
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绛旓細棣栧厛鎶婅繖涓悜閲忕粍鍖栦负琛屾渶绠褰㈠嵆闃舵鐭╅樀锛屾壘鍒版瘡鍒楅潪闆跺厓绱犲嵆鍙紝渚嬪锛歛1 a2 a3 a4 1 0 1 00 1 1 00 0 0 10 0 0 0 鏋佸ぇ绾挎ф棤鍏崇粍鍗充负锛歛1,a2,a4锛沘2,a3,a4锛沘1,a3,a4 a1,a2,a3涓嶆槸鏋佸ぇ鏃犲叧缁 ...
绛旓細a3 -a1 -2a2 = 0 a4 -2a1 -a2 = (0,0,-1,0)a5-2a1 -a2 +a4 -2a1 -a2 =0 鎵浠鏋佸ぇ绾挎ф棤鍏崇粍涓(a1,a2,a4),a3 =a1 +2a2 a5 = 4a1 +2a2 -a4
