Question

向量a1=(1,2,1)a2=(2,3,2)a3=(1,1,2)线性映射f:f(x,y,z)=A(x,y,z),f(a1)=a1，f(a2)=a1+a2，f(a3)=a2+2a3

求满足f(b)=λb的向量b和实数λ 题目中的向量都是列向量

Answer 1

设b=c1a1 + c2a2 + c3a3, 其中c1,c2,c3是参数，f(b)=c1*f(a1) + c2* f(a2) + c3*f(a3)
==>f(b)=(c1+c2)a1 + (c2+c3)a2 + 2c3a3 = λc1a1 + λc2a2 + λc3a3
==> c2 = (λ-1)c1 ; c3 = (λ-1)c2 ; (λ-2)c3 = 0;
当c3=0时，λ=1，带入后得c2=0,此时b=c1*a1,满足。
当λ=2时，c1=c2=c3,带入后满足，此时b=c(a1+a2+a3)