例8.13求向量组a1=2.1,4,3).a2=(-1,1,-6,6)3=(-1,-2,2,-9),a4=(1,1,-27),a6= (2,4,4,9)的一个极大线性无关组，并将其余向量用极大线性无关组线性表出. 例8.14(2006,34)设4维向量组a1=(1+a1,1,1,a2=(2,2+a2,27,a=3,3,3+a,3),a (44,4,4+aj,问a 为何值时a1,a2,,线性相关？当a,a2,a,a4线性相关时，求其 个极大线性无关 组，并将其余向量用极大线性无关组线性表出. 例8.15(2008)设a,B为3维列向量，矩阵A=aaT+3T,其中aT,8分别是a,3的转置.证明：(0秩r(A)≤ 2()若a,线性相关，则秩r(A)<2. 例8.16设A为m×n矩阵，证明：r(4TA)=r(4.~8.13 ¶ï˛|α1 = (2, 1, 4, 3), α2 = (−1, 1, −6, 6), α3 = (−1, −2, 2, −9), α4 = (1, 1, −2, 7), α5 = (2, 4, 4, 9)òá4åÇ5Ã'|,øÚŸ{ï˛^4åÇ5Ã'|Ç5L—. ~8.14 (2006,3,4) 4ëï˛|α1 = (1 + a, 1, 1, 1)T , α2 = (2, 2 + a, 2, 2)T , α3 = (3, 3, 3 + a, 3)T , α4 = (4, 4, 4, 4 + a) T ,Øa è¤äû,α1, α2, α3, α4Ç5É'?α1, α2, α3, α4Ç5É'û,¶Ÿòá4åÇ5Ã' |,øÚŸ{ï˛^4åÇ5Ã'|Ç5L—. ~8.15 (2008) α, βè3ëï˛,› A = ααT +ββT ,Ÿ•α T , βT©O¥α, β=ò.y²:(i) ùr(A) ≤ 2;(ii) eα, βÇ5É',Kùr(A) < 2. ~8.16 Aèm × n› , y²: r(AT A) = r(A). 9