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

## 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.

Last edited:

Ssnow
Gold Member
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
Gold Member
ok!