
Originally Posted by
jayshizwiz
(v1,v2 ,...,vk ,w) are vectors in vector space V
{ v1 − w,v2 − w,...,vk − w } spans V and w∉Sp{v1,v2 ,...,vk}
prove { v1,v2 ,...,vk } is dependant.
Attempt
I think it has something to do with this:
since { v1 − w,v2 − w,...,vk − w } - containing k vectors - spans V
that means (v1,v2 ,...,vk ,w) - containing k+1 vectors - are dependent
and maybe since w∉Sp{v1,v2 ,...,vk}
that means that in (v1,v2 ,...,vk ,w) w is not a linear combination of (v1,...,vk)
which then means that { v1,v2 ,...,vk } is dependent
What do you guys think??