How to define a given quotient space $\Bbb{R} ^2$/{$0$}?

Question

I am required to describe the following quotient space: $\Bbb{R} ^2/ \{0\}$.

Now, as far as I understand, a quotient space is a collection of sets such as $\{v_1 + U , v_2+U , \dots ,v_n + U\}$ where in this case every $v_i \in \Bbb{R}^2$ and $U$ is {$0$}.

So if I take $v_i$ as some vector $(a , b)$ and try finding out $v_1$ + {$0$}, I will have to evaluate $(a , b)$ + $0$ which makes no sense.

Can I say that the quotient space given is not possible?

Answer 1

Yes, it is possible. Your $\textbf{0}=(0,0)$. So $(a,b)+\{(0,0)\}=\{(a,b)\}$.

Answer 2

The quotient space $\mathbb{R}^2/\{ \mathbf{0} \}$ is well defined (where $\mathbf{0}= (0, 0) \in \mathbb{R}^2$). $$ \mathbb{R}^2/\{ \mathbf{0} \} := \{ \mathbf{a} + \mathbf{0}~:~ \mathbf{a} \in \mathbb{R}^2 \} $$ Moreover, a simple map will explain everything.

From the First Isomorphism theorem, for a vector space $V/\mathbb{K}$ and a subspace $W$ of $V$, we have the map $\Psi: V \to W$ such that $\Psi \in \mathcal{L}(V, W)$ and
$$ V/\text{ker}(\Psi) \cong \text{im}(\Psi) $$ From there we have that $$\mathbb{R}^2/\{\mathbf{0}\} \cong \mathbb{R}^2 $$