Question

设向量组A：a1,a2,a3线性无关，向量b1能由向量组A线性表示，向量b2不能由向量组A线性表示，k为任意常数，

问（1）向量组a1,a2,a3,kb1+b2是否线性相关，为什么？
（2）向量组a1,a2,a3,b1+kb2是否线性相关，为什么？

Answer 1

(1)
k1a1+k2a2+k3a3+k4(kb1+b2)=0
k1a1+k2a2+k3a3+k4(k(k1'a1+k2'a2+k3'a3)+b2)=0
(k1+kk1'k4)a1 +(k2+kk2'k4)a2+(k3+kk3'k4)a3+ k4b2=0
=>
(k1+kk1'k4) = 0 (1) and
(k2+kk2'k4) = 0 (2) and
(k3+kk3'k4) = 0 (3) and
k4=0 (4)
from (1),(2),(3),(4)
=> k1=k2=k3=k4=0
向量组a1,a2,a3,kb1+b2线性无关

(2)
k1a1+k2a2+k3a3+k4(b1+kb2)=0
k1a1+k2a2+k3a3+k4((k1'a1+k2'a2+k3'a3)+kk4b2)=0
(k1+k1'k4)a1 +(k2+k2'k4)a2+(k3+k3'k4)a3+ kk4b2=0
=>
(k1+k1'k4) = 0 (1) and
(k2+k2'k4) = 0 (2) and
(k3+k3'k4) = 0 (3) and
kk4=0 (4)
if k =0, 向量组a1,a2,a3,b1+kb2线性相关
if k不等于0，向量组a1,a2,a3,b1+kb2线性无关