设向量组a1,a2,a3,.,am与向量组a1,a2,.,am,b的秩相等,试证:l两向量组等价

let
很明显
向量组a1,a2,a3,.,am is a subset of 向量组a1,a2,.,am,b
因为 rank of {a1,a2,a3,.,am} = rank of {a1,a2,.,am,b}
b must be linearly combination of a1,a2,...,am
b = k1'a1+k2'a2+.+km'am
any linearly combination of a1,a2,.,am,b
= k1"a1+k2"a2+...+km"am+km+1"b
=k1"a1+k2"a2+...+km"am + k1'a1+k2'a2+.+km'am
=(k1"+k1')a1+ (k2"+k2')a1+...+(km"-km')am
is a linearly combination of {a1,a2,...,am}
=> 向量组a1,a2,.,am,b is a subset of 向量组a1,a2,.,am
=>向量组a1,a2,a3,.,am= 向量组a1,a2,.,am