dimension of intersection of subspaces, one of which of dimension $n-1$

Question

Let $H$ be a linear subspace of dimension $n-1$ of a linear space $V$ of dimension $n$.
Let $W$ be some subspace of $V$.
Show one of the following holds:

$W subseteq H$ or
$dim(W cap H) = dim(W) - 1$

This makes a lot of sense to me, as if there is some element in w that isn't in a largest subspace of V, then one of its dimensions can't be spanned by any of the vectors in the basis of $H$, thus its dimension is at most $dim(W) - 1$.
It is intuitive, but I can't come up with a proof...
please help

Sabino Di Trani · Accepted Answer

Hint: use the Grassman formula for dimension of $H + W$
$$dim (H+W)+dim (Hcap W)=dim (W) + dim (H)$$
If $W subset H$ then $H + W=H$, otherwise it is all the ambient space $V$.
Edit: suppose $H+W=V$, then $dim (H+W)=n$ and by Grassman formula:
$$n + dim (Hcap W)=dim (W) + dim (H)=dim (W) + n-1$$

dimension of intersection of subspaces, one of which of dimension $n-1$

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