Let $V$ be a vector space. What is quotient space $V/\{0\}$?

Question

How I understand this right now:

$V/\{0\} = \{v + \{0\} | \forall v \in V \}$.

So each coset is, fora any $v \in V$, $v +\{0\} = \{v+0\} = \{v\}$.

So, $V/\{0\}=\{\{v\}|\forall v \in V\}$, which is $\textbf{not}$ the same as $V$ since every element of $V/\{0\}$ is a set containing a single element of $V$. But my textbook says $V/\{0\}=V$.

I have seen the complement problem, Why isn't the quotient space $V/V = \{ V \}$?

but I'm still not understanding something...

Please help me understand what I'm confused about.

Answer 1

The map $v\rightarrow\{v\}$ is isomorphic as vector spaces: $\{v\}+\{w\}=\{v+w\}$, $k\{v\}=\{kv\}$, and if $\{v\}=\{0\}$, then $v=0$.

So we identify $V$ and $V/\{0\}$.

Answer 2

Well! Your book uses abuse of language, it's obviously unequal but since it is Isomorphic then we can say they are same structurally. Hope it works