Intersection of subspaces: Does there exist for a subspace a non trivial subspace such that that their intersection has dimension 1?

Question

More specifically: Let $V$ be an $n$ dimensional vector space over a finite field $F_{q}$. Let $S\subset V$ be a $k$-dimensional ($k> 1$) subspace. Does there exist an $m$-dimensional subspace $W\subset V$, with $m=\dim(W)>1$ such that $\dim(S\cap W)=1$?

Answer 1

Hint. Let $T$ be the span of $S$ and $W$. Then $$ \dim(T) = \dim(S) + \dim(W) - \dim(S \cap W) $$

Whenever you can satisfy this equality there will be such a $W$.

The fact that the field is finite is irrelevant.

Answer 2

it is impossible because of the assumption which is $W\subset V$

$W\subset V \Rightarrow W \cap V=W$

$W \cap V=W \Rightarrow dim(W \cap V)=dim(W)>1$ which means $dim(W) \neq1$

Answer 3

Given a vector basis of V, and considering the subbasis that spans the subspace $S$ (S is not all V) then it is enough to take the basis of V, and substract all the vectors except 1 of those that span $S$ (considering the subbase for $S$ formed by vectors of the base for $V$). Then the resulting spanned subspace is the $W$ i have been looking for. It is clear that $W$ is not unique.