Let $V$ a vector space with $\dim V < \infty$ and $U$ a subspace of $V$. Let $V/U$ a factor space, i.e. $V/U:=\{[x]\mid x\in U\}$ with $[x]:=\{y\in V\mid x-y \in U\}$
I have four elementary questions.
$\bullet_1$ If $v,w\in V$. Is it true that $v\ne w$ iff $[v]\ne[w]$ ?
$\bullet_2 $ If $U=\{0\}$, then $V/U=P(V)$? ($P$ denotes the potence set)
$\bullet_3$ If $U=V$, then $V/U=\{V\}$ or $V/U=V$ ?
$\bullet_4$ $\{(x,y)\in V\times V\mid x+y\in U\}$ is a equivalence relation on $V$?
My thoughts:
According to $\bullet_1$: $'\Rightarrow'$ holds, but not $'\Leftarrow'$.
According to $\bullet_2$:, I think this is true since $[x]=\{0\}$ and $\{0\}\in W$ for any $W\subset P(V)$
According to $\bullet_3$: I would say $V/U=V$, since $\{V\}$ is not even a vector space.
According to $\bullet_4$: I know that $\{(x,y)\in V\times V\mid x-y\in U\}$ is a equivalence relation on $V$. And since $y\in V$ also $-y\in V$, thus this is the same?
I am absolutely not sure about factor spaces. Some enlightenment would be very helpful!
Ad 1: $\Leftarrow$ certainly holds, $\Rightarrow$ in general not. (But if $U=\{0\}$, ...)
Ad 2: In general, no. Instead $V/U=\{\,\{x\}\mid x\in V\,\}$. And as soon $V$ is not the 0-space, we can find distinct elements $x,y\in V$ and have $\{x,y\}\in P(V)$, but $\{x,y\}\notin V/U$.
Ad 3: $V/U=\{V\}$ is a one element set. Which is a zero-dimensional vector space.
Ad 4: In general, this is not an equivalence relation. For an equivalence relation, you'd need (among others) that it contains $(v,v)$ for all $v\in V$. But you cannot expect $2v\in U$ to hold for all $v\in V$. This is only true if either $U=V$ or we work in characteristic $2$.