Why is $V/W$ all multiples of $(0,1)^T$ given $V = \mathbb{R}^2$ and $W$ is all multiples of $(1,0)^T$?

Question

Definitions: $v+ U \equiv \left \{v +u: u \in U \right \}$ and $V/U \equiv \left \{v + U: v \in V \right \}$

With these I figured that in this example $v+ W \in V/W$ would be some line parallell to the $x_1$-axis, and thus $V/W$ is all lines parallell to the $x_1$-axis.

Answer 1

$V/W$ is the set of all lines, parallel to the $x$ axis.

However, as a vector space, $V/W$ is isomorphic to the vertical axis, as the mapping

$$(0, y)\mapsto \{(x, y)|x\in\mathbb R\}$$

is an isomorphism.

Note also that this means that $V/W$ is isomorphic to $\mathbb R$.