已知n维向量a1,a2,a3,a4,a5线性无关,A是n阶可逆矩阵,证明Aa1,Aa2,Aa3,Aa4,Aa5线

题目

已知n维向量a1,a2,a3,a4,a5线性无关,A是n阶可逆矩阵,证明Aa1,Aa2,Aa3,Aa4,Aa5线

答案

因为 (Aa1,Aa2,Aa3,Aa4,Aa5) = A(a1,a2,a3,a4,a5)
且A可逆
所以 r(Aa1,Aa2,Aa3,Aa4,Aa5) =r[ A(a1,a2,a3,a4,a5)] = r(a1,a2,a3,a4,a5) = 5
所以 Aa1,Aa2,Aa3,Aa4,Aa5 线性无关.