Why isn't the quotient space $V/V = \{ V \}$?

Question

If $W \subset V$, then one defines the quotient space, $$V/W = \{ v + W : v \in V \}$$

So why isn't this right?

$$V/V = \{v + V : v \in V \} = \{V \}$$?

I read that $V/V = \{ 0 \}$? Why can't the whole set $V/V$ be partition, by $V$ itself?

The coset $V=0+V$ is also the zero element $0=0_{V/V}$ of the quotient space $V/V$. In other words, you are right, and so is the person claiming that $V/V={0}$. — Jyrki Lahtonen, Oct 12 '14 at 22:29

Belgi · Answer 1 · 2014-10-12T22:38:30.653

7

$V/W$ is a vector space, and it have the zero element $0_{V/W}$.

I claim that $0_{V/W}=W$ - Indeed if $v+V\in V/W$ then $$ (v+V)+W=W+(v+V)=v+V $$

so $$ \{V\}=\{0_{V/V}\} $$

that is - it is the zero subspace.

edited Oct 12 '14 at 22:38

answered Oct 12 '14 at 22:30

Belgi

I guess your claim is rather $0_{V/V}=V$ (which must be the case as $V/V={V}$ has only one element). – Berci Oct 12 '14 at 22:36
Oh okay I understand what they mean now. Thanks. – Lemon Oct 12 '14 at 22:37
@Berci - thanks, I worked with $W=V$. I edited accordingly – Belgi Oct 12 '14 at 22:39

score 7 · Accepted Answer · answered Oct 12 '14 at 22:31

7

You are right, $V/V=\{V\}$. The reason this is sometimes written $\{0\}$ is that $V$ functions as the zero vector in $V/V$, i.e., in $V/V$, $V=0$.

answered Oct 12 '14 at 22:31

Mike Earnest

Oh, okay I forget that. The notes I read online just writes "$0$", so that's what they meant. – Lemon Oct 12 '14 at 22:36

2 Answers2