How to show that $Sp\{u+v-3w,2v-w,t+w,v+w\} = Sp\{u,v,w,t\}$

Question

I know how to show that left part of the equation is included in the right part - I show that for $a,b,c,d \in \mathbb R$ it equals to $au+(a-b)v+(-3a-b+c+d)w-ct$, hence all vectors of the left span are included in the right span. But how to show the opposite direction? Thanks.

Answer 1

Hint:

Construct four linear equations and check if a solution exist, one equation is $$ u=\alpha(u+v-3w)+\beta(2v-w)+\gamma(t+w)+\delta(v+w) $$

Answer 2

The left hand side contains $(2v-w)+(v+w)=3v$, so if $3\ne 0$ it contains $v$. Using that, it also contains $(v+w)-v=w$, then $(t+w)-w=t$ and $(u+v-3w)-v+3w=u$.