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

Mathematics Asked by Gulzar on July 27, 2020

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:

1. $$W subseteq H$$ or
2. $$dim(W cap H) = dim(W) – 1$$

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…

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:

$$n + dim (Hcap W)=dim (W) + dim (H)=dim (W) + n-1$$

Correct answer by Sabino Di Trani on July 27, 2020

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