a1a2是非其次线性方程ax=b的两个不等解怎么证a1,a1-a2线性无关啊

Question

a1 a2是非其次线性方程ax=b的两个不等解 怎么证a1,a1-a2线性无关啊

Answer 1

设实数m,n满足
m(a1+e)+n(a2+e)=0,(1)
则A[m(a1+e)+n(a2+e)]
=m(Aa1+Ae)+n(Aa2+Ae)
=mb+nb+
=（m+n+)b=0,
b≠0,
∴m+n+=0,
代入（1）,得
ma1+na2+=0,
a1,a2,线性无关,
a1,a1-a2线性无关.