So I have a set of 2 vectors, $v1,v2$ which are linearly independent and the dimension of the subspace they belong to is 3 ($dim(W) =3)$. How can I prove there is some vector $W$ that is not a linear combination of $v1,v2$?
edit: my attempt so far, I assumed that some vector $W$ is in $span(v1,v2)$
I show some vector in the subspace, $W$ can be written as a linear combination of $v1,v2$
then i showed $v1 + v2 -W = 0$ and hence $W,v1,v2$ and linearly dependent which contradicts the original assumption that $W$ is in $span(v1,v2)$