Why is this quotient space the zero space?

Question

The text says “The quotient space $V/V$ is the zero space as $x+V=V$ for all $x \in V$. I can only seem to think of $V/V = \{V\}$, since the coset $x+V$ (with $x \in V$) always represents the set $V$ itself.

Answer 1

You are right: $V/V$ consists of a single element, which is $V$. And the only vector space with a single element is the zero space.

Besides, note that, in $V/V$,\begin{align}V+V&=(0+V)+(0+V)\\&=(0+0)+V\\&=0+V\\&=V.\end{align}So, $V$ is the null vector of $V/V$ (as it would be expected).