May $V$ be an $n$ dimensional Vektorspace such that $\dim (V) =: n \ge 2$.
We shall prove, that there are infinitely many $k$-dimensional subspaces of $V$, $\forall k \in \{1, 2, …, n-1\}$.
So first, I thought about using induction, the base step is not that hard, for $n=2$ we take two vectors, say $a$ and $b$ and define infinitely many 1-dimensional subspaces as span$\{a+jb\}$ for $j \in \mathbb N$.
It is easy to see those vector spaces are not all equal, but I kinda realised that induction is not the way to go, as I think $n$ is fixed.
Anyhow, then I thought about using finiteness of basis for $V$ to try to construct those subspaces (using vectors from basis). I failed to do so, so I'm just asking for a hint or any useful advice where to start with this.
Best Answer
Subspaces:$(a_1\times a_2\times...\times a_{k-1}\times(a_k+ma_{k+1}))$, $m\in N$