# Math Help - Vector space and maximal proper subspace

1. ## Vector space and maximal proper subspace

Suppose that V is a vector space over F and $\vec{v} \in V$ is any nonzero vector.
Show that there is a subspace W of V maximal subject to the condition
that $\vec{v} \notin W$ ; show that any such W is a maximal proper subspace of V .
(Do NOT assume that V is ﬁnite-dimensional.)
For the first part I thought that I could say :
1. Prove that we can exchange one of the vector in basis of V by $\vec{v}$.
2. Prove that the basis (B) is still one.
3.Let $W = B \backslash \vec{v}$.
4. $B \backslash \vec{v}$ is a basis and thus a maximal independant subset that doesn't contain $\vec{v}$.
5. W is a subspace of V.
Is it correct?
For the second part I don't have a good idea of what to do.
Any hint would be great. Thanks!

2. Originally Posted by vincisonfire
Suppose that V is a vector space over F and $\vec{v} \in V$ is any nonzero vector.
Show that there is a subspace W of V maximal subject to the condition
that $\vec{v} \notin W$ ; show that any such W is a maximal proper subspace of V .
(Do NOT assume that V is ﬁnite-dimensional.)
the first part is a quick result of Zorn's lemma: let $C$ be the set of all subspaces W of V such that $v \notin W.$ since $v \neq 0,$ the set $C$ contains the 0 subspace and hence it's not empty.

choose any chain (totally ordered collection) of elements of $(C, \subseteq).$ then the union of those elements is still in $C.$ thus $C$ has a maximal element by Zorn's lemma.

for the second part, suppose $W$ is a maximal element of $C$ and $W \subsetneq W_1,$ for some subspace $W_1$ of $V.$ the claim is that $W_1=V$: by maximality of $W$ in $C,$ we must have $v \in W_1.$

thus $W+ \subseteq W_1.$ so we only need to show that $W+=V.$ suppose $W+ \neq V,$ and let $v_1 \notin W + .$ then $v \notin W + $ and thus $W + \in C,$ which

contradicts maximality of $W$ in $C. \ \Box$

3. The version of Zorn's lemma I know says that $

C
$
needs to be closed under union of chains.
Here it is
Let U be any set, and B any nonempty collection of subsets of U. Suppose tnat B is closed under unions of chains, then B has a maximal element.

How is define the union of these chains?

4. Originally Posted by vincisonfire

How is define the union of these chains?
if $\{W_{\alpha}: \ \alpha \in I \}$ is a chain of subspaces (under inclusion), then $\bigcup_{\alpha \in I} W_{\alpha}$ is again a subspace. if $\forall \alpha \in I: \ W_{\alpha} \in C,$ as defined in my solution, then $\bigcup_{\alpha \in I} W_{\alpha} \in C.$