Equivalence relation on V

Question

Trying to prove equivalence relation on $V$. $v\sim w$ if there is $u\in U$ such that $v=w+u$. I know that we can prove $\sim$ is an equivalence relation on $V$ by showing it is reflexive, symmetric and transitive.

So far I have:

(reflexive) $v = v+0$ for all $v\in V$ so $v\sim v$ and it is transitive.

(symmetric) $v=w+u, \quad u=(-w)+v$ so $w\sim v$ and it is symmetric.

(transitive) $v=u_{1}+w$ and $w=u_{2}+z$

$v+w=u_{1}+u_{2}+w+z$

$v=u_{1}+u_{2}+x$

so $v\sim z$ and it is transitive. Am I doing it right?

Answer 1

Suppose $u \sim v$ and $v \sim w$. Then, $u=v+x$ and $v=w+y$ for some vectors $x$ and $y$ on $\textsf U$. Thus $$u=(w+y)+x=w+(y+x)$$ so, $u \sim w$ since $y+x\in \textsf U$ (assuming $\textsf U$ is a vector subspace of $\textsf V$).

Answer 2

Assuming that $V$ is a vector space, and $U$ is assumed to be a subspace of $V$, your proof has the right gist, but it's a bit sloppy. The first thing should be something like:

(reflexive) We need to show that for all $v \in V$, there's a $u \in U$ with $v+u = v$. Because $U$ is a subspace, and therefore contains the 0-vector, we can choose $u = 0$ and get $v + u = v + 0 = v$.

I grant you that's a bit wordy, but it does state explicitly what needs to be shown, and then explains why $0$ is allowed as a possible "u" in the proof.

For the third one, I'll get you started:

(transitive) Suppose that $v \sim w$ and $w \sim x$; we need to show that $v \sim x$, i.e., that there's a vector $u \in U$ with $v + u = x$. From the first assumption, we know there's a vector $u_1 \in U$ with $v + u_1 = w$. From the second...

Of course, both of these are proofs suitable for someone who's just learning linear algebra. In a research paper, it would probably suffice to say something like "Because $U$ is a subspace of $V$, the relation $v \sim w$ iff $v-w \in U$ is evidently an equivalence relation." So there is (as usual in mathematics) a question of audience for any given proof.

Answer 3

Symmetry and transitivity are not correct.

You need that $U$ is a subspace of $V$.

(1) $v\sim v$, since $v=v+0$ with $0\in U$.

(2) If $v\sim w$, i.e., $v=w+u$ for some $u\in U$, then $w = v+(-u)$ with $-u\in U$, i.e., $w\sim v$.

(3) If $v\sim w$ and $w\sim x$, i.e., $v=w+u$ and $w=x+u'$ for some $u,u'\in U$, then $x= w - u' = v - u - u' = v+(-u-u')$ with $-u-u'\in U$, i.e., $v\sim x$.