.数学帝及海外党进Let S = {向量a,向量b} 是 spanning set of R2.For what real values of t s.t S ={向量a+t*向量b}也是spanning set of R2?给出证明原题

Question

.数学帝及海外党进
Let S = {向量a,向量b} 是 spanning set of R2.For what real values of t s.t S ={向量a+t*向量b}也是spanning set of R2?
给出证明

原题

Answer 1

a,a+tb 满足要求的充要条件是两个向量组 a,a+tb 与 a,b 等价
显然 a,a+tb 可由 a,b 线性表示,所以只需满足 R(a,a+tb) = R(a,b) = 2
即需满足 a,a+tb 线性无关
(a,a+tb) = (a,b)K
K =
1 1
0 t
由于 a,b 线性无关,所以 R(K) = R(a,a+tb) = 2.
所以 t ≠ 0.