Prove :Union of Three subspaces is a subspace if ....

[Saph]

Homework Statement

Prove the the union of three subspaces is a subspace if one of the subspaces contains the others

Homework Equations

A subset W of a vector space V is called a subspace if : 1) $0 \in W$. 2) if $U_1$ and $U_2$ are in $W$, then
$U_1 + U_2 \in W$, 3) if $\alpha$ is a scalar, then $\alpha U\in W$

The Attempt at a Solution

assume that $\exists~x,y,z \in U_1\cup U_2\cup U_3 ~$ such that, $x \in U_1 ~, y \in U_2 ~ and~~ z \in U_3$.
We know that, $x+y+z~\in U_1\cup U_2\cup U_3$, hence $x+y+z~is~in~either~U_1 ~or~U_2 ~or ~U_3$
Assume, WOLOG, that $x+y+z~\in~U_1 ,~then~ y+z \in U_1 ,~moreover,~y+z\in U_1 \cup U_2~$,thus
$z\in U_1 \cup U_2~,~and~we~have~two~cases~to~consider$.
$i)~ z \in U_1 ~,~then~y+z\in U_1 ,~\implies~y\in U_1 ~, thus,~ any~z\in~U_3 ~, then~z\in U_1~,~and~any~y \in~U_2 ~, then~y\in U_2$
$hence,~U_2 ~and~U_3 ~\subset U_1$
$ii) ~z\in U_2$, then I don't know how to proceed.

 

[Ssnow]

Hi, you can start considering two. Assume that $U$ and $W$ are subspaces of a vector space $V$. It is possible to prove that if $U\cup W$ is a subspace then either $U\subseteq W$ or $W\subseteq U$. The idea is that: assume $U\not\subseteq W$ and $W\not\subseteq U$ and pick $u\in U$ and $w\in W$ with $u\not\in W$ and $w\not\in U$ then look at the sum $u+w\in U\cup W$?

 

[Saph]

Hi, you can start considering two. Assume that $U$ and $W$ are subspaces of a vector space $V$. It is possible to prove that if $U\cup W$ is a subspace then either $U\subseteq W$ or $W\subseteq U$. The idea is that: assume $U\not\subseteq W$ and $W\not\subseteq U$ and pick $u\in U$ and $w\in W$ with $u\not\in W$ and $w\not\in U$ then look at the sum $u+w\in U\cup W$?

Hello, thank you for your answer, I have proved the case for the union of two subspaces (I used the same idea that you suggested ) , my problem is the union of three subspaces.
The post is not complete yet, as I'am learning how to post using latex,I intended to delete this thread but I couldn't, so right now I'am editing the thread to fix latex problems and include my proposed answer.

 

[Ssnow]

ok!