$W$ intersection of $(n-1)$ dimensional subspaces

Question

I have got a good (I think so) intuition of this problem but I am not being able to write down the crucial steps correctly.

Let $V$ be a $n$ dimensional vector space over field $F$ . Let $W$ be a subspace of $V$ with dimension $p\lt n$. Then show that $W$ is the intersection of all $(n-1)$ dimensional subspaces of $V$ which contains $W$.

My intuition : Suppose we are considering integers from 1 to 10. ( I am not saying {1 to 10} is a vector space, I am just giving an analog ). And my $'W'$ be the number $2$. I say that $2$ is the intersection of the factors of all numbers from 1 to 10 which are multiples of 2.

Answer 1

If $U$ is a subspace of $V$, then we call $\dim V-\dim U$ the codimension of $U$ in $V$. Thus your problem is naturally phrased as

Show that $W$ is the intersection of all codimension one subspaces of $V$ containing $W$.

Here's an outline of the proof:

Extend a basis $\{w_1,\dotsc,w_p\}$ of $W$ to a basis $\{w_1,\dotsc,w_p,v_1,\dotsc,v_{n-p}\}$ of $V$. For $1\leq j\leq n-p$ let $$ H_j=\DeclareMathOperator{Span}{Span}\Span\bigl(\{w_1,\dotsc,w_p,v_1,\dotsc,v_{n-p}\}\setminus\{v_{j}\}\bigr) $$ Note that each $H_j$ is a codimension one subspace of $V$ containing $W$. Moreover we can prove that $$ W=H_1\cap \dotsb \cap H_{n-p}\tag{1} $$ Can you prove (1)?

Once (1) has been established the result is quite easy. Let $\cal U$ be the collection of codimension one subspaces of $V$ containing $W$. Then $$ W\subseteq\bigcap_{U\in\cal U} U\subseteq H_1\cap\dotsb\cap H_{n-p}=W $$ Hence $$ W= \bigcap_{U\in\cal U} U $$ as desired.