线性代数：最大无关向量表示问题
\u7ebf\u6027\u4ee3\u6570\u6700\u5927\u5411\u91cf\u65e0\u5173\u7ec4\u6709\u54ea\u4e9b\u5411\u91cf\u600e\u6837\u786e\u5b9a(a1,a2,a3,a4) \u7ecf\u521d\u7b49\u884c\u53d8\u6362\u5316\u4e3a\u68af\u77e9\u9635
\u975e\u96f6\u884c\u7684\u9996\u975e\u96f6\u5143\u6240\u5728\u5217\u5bf9\u5e94\u7684\u5411\u91cf, \u5373\u6784\u6210\u4e00\u4e2a\u6781\u5927\u65e0\u5173\u7ec4
\u5982 (a1,a2,a3,a4,a5) \u5316\u4e3a
1 2 3 4 5
0 0 6 7 8
0 0 0 0 9
0 0 0 0 0
a1,a3,a5 \u4e3a\u4e00\u4e2a\u6781\u5927\u65e0\u5173\u7ec4
\u53ea\u8981\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4x1a1+x2a2+x3a4=a3 \u5373\u53ef\u3002
x1\uff0cx2\uff0cx3\u6c42\u51fa\u6765\u4e86\uff0ca3\u81ea\u7136\u5c31\u7528a1\uff0ca2\uff0ca4\u8868\u793a\u51fa\u6765\u4e86.
绛旓細杩欎釜鏄剧劧璇ョ煩闃电殑绉╀负4.杩欐椂鍊欐樉鐒舵瀬澶绾挎ф棤鍏缁勬槸杩欏洓涓鍚戦噺銆傚洜姝や换浣曚竴涓悜閲忎笉鑳界敤鍙﹀涓変釜鏉ヨ〃绀鸿屽繀椤荤敤鍥涗釜鍚戦噺琛ㄧず銆傚鏋滅湡瑕佽〃绀何4锛岄偅鍙兘琛ㄧず鎴愇4=0*伪1+0*伪2+0*伪3+伪4
绛旓細鏈澶ф棤鍏缁勬槸绾挎т唬鏁涓殑涓涓蹇碉紝瀹冩槸鎸囦竴涓鍚戦噺缁勪腑鏈澶х殑绾挎ф棤鍏冲瓙闆嗐傚湪琛ㄧず浣欏悜閲忔椂锛屾垜浠彲浠ヤ娇鐢ㄦ渶澶ф棤鍏崇粍鏉ョ畝鍖闂銆傞鍏堬紝鎴戜滑闇瑕佹壘鍒颁竴涓悜閲忕粍鐨勬渶澶ф棤鍏崇粍銆傝繖鍙互閫氳繃楂樻柉娑堝厓娉曟垨鑰呮牸鎷夊-鏂藉瘑鐗规浜ゅ寲杩囩▼鏉ュ疄鐜般傚叿浣撴楠ゅ涓嬶細1.灏嗗悜閲忕粍鍐欐垚鐭╅樀鐨勫舰寮忥紝姣忎竴鍒椾唬琛ㄤ竴涓悜閲忋
绛旓細1銆鏋佸ぇ绾挎ф棤鍏缁勶紙maximal linearly independent system锛夋槸绾挎х┖闂寸殑鍩哄鍚戦噺闆嗙殑鎺ㄥ箍銆傚叾瀹氫箟涓猴細璁維鏄竴涓猲缁村悜閲忕粍锛屛1,伪2,...伪r 鏄疭鐨勪竴涓儴鍒嗙粍锛屽鏋滄弧瓒 (1) 伪1,伪2,...伪r 绾挎ф棤鍏筹紱(2) 鍚戦噺缁凷涓瘡涓涓悜閲忓潎鍙敱姝ら儴鍒嗙粍绾挎ц〃绀锛岄偅涔埼1,伪2,...伪r 绉颁负鍚戦噺...
绛旓細鍒 a3 = -a1-a2, a5 = 4a1+3a2-3a4
绛旓細绾挎т唬鏁鏄唬鏁板鐨勪竴涓垎鏀紝涓昏澶勭悊绾挎у叧绯闂銆傜嚎鎬у叧绯绘剰鍗虫暟瀛﹀璞′箣闂寸殑鍏崇郴鏄互涓娆″舰寮忔潵琛ㄨ揪鐨勩備緥濡傦紝鍦ㄨВ鏋愬嚑浣曢噷锛屽钩闈笂鐩寸嚎鐨勬柟绋嬫槸浜屽厓涓娆℃柟绋嬶紱绌洪棿骞抽潰鐨勬柟绋嬫槸涓夊厓涓娆℃柟绋嬶紝鑰岀┖闂寸洿绾胯涓轰袱涓钩闈㈢浉浜わ紝鐢变袱涓笁鍏冧竴娆℃柟绋嬫墍缁勬垚鐨勬柟绋嬬粍鏉琛ㄧず銆傚惈鏈塶涓湭鐭ラ噺鐨勪竴娆℃柟绋嬬О涓虹嚎鎬...
绛旓細绾挎т唬鏁鏄竴闂ㄩ噸瑕佺殑璇剧▼锛屽濂藉畠鍙互瑙ｅ喅鐢熸椿涓殑寰堝闂锛屼粖澶╀粙缁嶇殑灏辨槸绉╁拰鏈澶ф棤鍏缁勩1銆侀鍏堜粙缁嶄竴涓嬬煩闃电殑绉╃殑姒傚康銆傝瀹氾紝闆堕樀鐨勭З涓洪浂锛屽彲閫嗛樀鍙堢О涓烘弧绉╂柟闃.2銆佷簡瑙ｅ畬鐭╅樀鐨勭З锛屽啀浜嗚В涓涓嬶紝鍚戦噺缁勭殑绉┿傚凡鐭ュ悜閲忕粍 A :a1, a2, 鈥, an 锛岃嫢A 鐨勪竴涓儴鍒嗙粍A0 锛歛1,a2, 鈥...
绛旓細灏辨槸绗3鍒=2脳绗1鍒-绗1鍒椼傝繖涓鍚戦噺缁勪腑鐨勪换浣曚竴涓悜閲忛兘鍙互鐢ㄧ1鍒椼佺2鍒椼佺3鍒绾挎琛ㄥ嚭銆
绛旓細娌℃湁浠讳綍闂銆傚洜涓哄悜閲忕┖闂寸殑鏈澶х嚎鎬ф棤鍏冲悜閲缁勭殑瀹氫箟灏辨槸锛1.鍙互琛ㄧず鍚戦噺绌洪棿鍐呴儴鎵鏈夊悜閲忥紱2.澶氫竴涓悜閲忓嵆绾挎х浉鍏炽
绛旓細鍚戦噺缁勭殑绉┿鏈澶ф棤鍏缁勭殑姒傚康鍙婂叾璁＄畻鏂规硶濡備笅锛氬湪绾挎т唬鏁涓紝鍚戦噺缁勭殑绉╁拰鏈澶ф棤鍏崇粍鏄潪甯搁噸瑕佺殑姒傚康锛屽畠浠湪鐭╅樀杩愮畻鍜岀嚎鎬ф柟绋嬬粍鐨勬眰瑙ｄ腑璧风潃鑷冲叧閲嶈鐨勪綔鐢ㄣ傛湰鏂囧皢瀵瑰悜閲忕粍鐨勭З銆佹渶澶ф棤鍏崇粍鐨勬蹇佃繘琛岃缁嗕粙缁嶏紝骞舵帰璁ㄥ畠浠殑璁＄畻鏂规硶銆傞鍏堬紝璁╂垜浠潵浜嗚В涓涓嬪悜閲忕粍鐨勭З鏄粈涔堟剰鎬濄傚悜閲忕粍鐨...
绛旓細鍏堝皢鍚戦噺鏋勬垚鐨勭煩闃电敤鍒濈瓑琛屽彉鎹㈠寲涓鸿鏈绠褰 1 0 0 -1 0 1 0 -1 0 0 1 1 0 0 0 0 鐪嬫渶鍚庝竴鍒楃殑闈為浂鍏冪礌涓庡墠闈㈠嚑涓"1" 鐨勫叧绯 鍋囧鍚戦噺鐨勯『搴忔槸a1,a2,a3,a4 鍒 a4 = -a1-a2+a3.
