求向量组的秩及一个极大无关组，并把其余向量用极大无关组线性表示。
【答案】：极大无关组为：α1，α2，α4，且α3=3α1+α2，α5=-α1-α2+α4$α2，α3，α4为极大无关组α1=α2+α3+α4绛旓細0 1 2 1 0 0 -3 -1 鎵浠鍚戦噺缁勭殑绉涓3, 伪1,伪2,伪3鏄涓涓瀬澶ф棤鍏崇粍.[娉:1.涓嶇鍚戦噺缁勭粰鐨勬槸琛屽悜閲忚繕鏄垪鍚戦噺, 鏋勯犵煩闃垫椂閮借浆鍖栨垚鍒楀悜閲 鐒跺悗杩涜鍒濈瓑琛屽彉鎹.2.杩欑被棰樼洰鍖栨垚姊舰灏卞彲浠ヤ簡.闈為浂琛屾暟灏辨槸鍚戦噺缁勭殑绉.闈為浂琛岀殑棣栭潪闆跺厓鎵鍦ㄥ垪瀵瑰簲鐨勫悜閲,灏辨槸涓涓瀬澶ф棤鍏崇粍....
绛旓細杩涜鍒濈瓑琛屽彉鎹㈠嵆鍙 1 0 3 1 -1 3 0 -1 4 2 14 0 r2+r1锛宺3-4r1 ~1 0 3 1 0 3 3 0 0 2 2 -4 r2/3锛宺3-2r2 ~1 0 3 1 0 1 1 0 0 0 0 -4 r3/-4锛宺1-r3 ~1 0 3 0 0 1 1 0 0 0 0 1 浜庢槸鍚戦噺缁勭殑绉涓3 鑰屽悜閲忕粍鐨鏋佸ぇ鏃犲叧缁鏄痑1锛宎2锛宎4 ...
绛旓細鍖栨垚鍒楅樁姊舰鐭╅樀锛岃繃绋嬪鍥俱傜敱(2)寮忓彲寰楃煩闃礎鐨勭З涓3锛屽嵆鍘鍚戦噺缁勭殑绉涓3銆(1)寮忕殑绗簩鍒椼佺涓夊垪鏄垚姣斾緥鐨勶紝鍥犳瀵瑰簲鐨勎2銆佄3鏄彲浠ヤ簰鐩歌〃绀虹殑銆傚洜姝ら夊彇鏋佸ぇ绾挎ф棤鍏崇粍鏃讹紝伪1銆佄4蹇呴夛紝伪2銆佄3浜岄変竴銆傛晠鏋佸ぇ鏃犲叧缁涓簕伪1锛屛2锛屛4}鎴杮伪1锛屛3锛屛4}銆
绛旓細r2+r1,r4-4r1 1 2 1 3 0 3 0 3 0 5 -2 7 0 -2 -4 2 r2*(1/3),r1-2r2,r3-5r2,r4+2r2 1 0 1 1 0 1 0 1 0 0 -2 2 0 0 -4 4 r3*(-1/2),r1-r3,r4+4r3 1 0 0 2 0 1 0 1 0 0 1 -1 0 0 0 0 鎵浠,a1,a2,a3 鏄涓涓瀬澶ф棤鍏崇粍 a4=2a1+...
绛旓細12-2k 0 0 0 -4 锛夛紱棰樻剰寰桝涓轰笉鍙嗙煩闃碉紝鎵浠AI=0锛1*锛-4锛*锛12-4k锛*锛-4锛=0 k=3 褰揔=3鏃讹紝r锛圓锛=3锛孉= 锛 1 2 0 2 0 -4 -4 -2 0 0 0 6 0 0 0 -4 锛 A=锛坅1,a2,a3,a4锛鍒欐瀬澶ф棤鍏崇粍涓築=锛坅1锛宎2,a4锛 a3=(-2)*a1+1*a2+0*a4銆
绛旓細[1 1 0 1][0 1 -1 2][0 0 1 -1][0 0 0 0]鍒欏悜閲忕粍鐨勭З涓3锛宎1锛宎2锛宎3 涓涓涓瀬澶绾挎鏃犲叧缁銆傚啀琛屽垵绛夊彉鎹负 [1 1 0 1][0 1 0 1][0 0 1 -1][0 0 0 0]琛屽垵绛夊彉鎹负 [1 0 ...
绛旓細鍒濈瓑琛屽彉鎹负 [1 0 0 -3][0 1 1 4][0 0 -1 1]鍒濈瓑琛屽彉鎹负 [1 0 0 -3][0 1 0 5][0 0 1 -1]a1,a2,a3 涓涓涓瀬澶绾挎鏃犲叧缁銆俛4 = -3a1+5a2-a3
绛旓細0 0 0 -4 -4 r2-3r3 1 0 3 2 1 0 0 0 0 0 0 1 1 1 0 0 0 0 -4 -4 r4*(-1/4),r1-2r4,r3-r4 1 0 3 0 -1 0 0 0 0 0 0 1 1 0 -1 0 0 0 1 1 鎵浠鍚戦噺缁勭殑绉涓3, a1,a2,a4 鏄涓涓瀬澶ф棤鍏崇粍 涓 a3...
绛旓細12-2k 0 0 0 -4 锛夛紱棰樻剰寰桝涓轰笉鍙嗙煩闃碉紝鎵浠AI=0锛1*锛-4锛*锛12-4k锛*锛-4锛=0 k=3 褰揔=3鏃讹紝r锛圓锛=3锛孉= 锛 1 2 0 2 0 -4 -4 -2 0 0 0 6 0 0 0 -4 锛 A=锛坅1,a2,a3,a4锛鍒欐瀬澶ф棤鍏崇粍涓築=锛坅1锛宎2,a4锛 a3=(-2)*a1+1*a2+0*a4銆
绛旓細瑙: (a1,a2,a3,a4) = -1 1 6 -2 1 3 2 4 1 5 6 5 r3-r2,r2+r1 -1 1 6 -2 0 4 8 2 0 2 4 1 r2-2r3 -1 1 6 -2 0 0 0 0 0 2 4 1 鎵浠鍚戦噺缁勭殑绉涓2, a1,a2 鏄涓涓瀬澶ф棤鍏崇粍.
