Proof verification: $ \overline{v}_{1} = \overline{v}_{2} \iff v_{1} - v_{2} \in W \iff v_{2} - v_{1} \in W.$

Question

Let W be a subspace of V. Then for $\overline{\textbf{v}}_{i} = \overline{\textbf{v}}_{i} + W $, $$ \overline{\textbf{v}}_{1} = \overline{\textbf{v}}_{2} \iff \textbf{v}_{1} - \textbf{v}_{2} \in W \iff \textbf{v}_{2} - \textbf{v}_{1} \in W.$$

I need some verification on my proof:

$\Rightarrow$ Let $ x \in \mathbf{v}_{1} + W$ then $x = \mathbf{v}_{1} + \mathbf{w}_{1}$ for some $\mathbf{w}_{1} \in W$.

$x \in \mathbf{v}_{2} + W$ thus $ x = \mathbf{v}_{2} + \mathbf{w}_{2}$

$\mathbf{v}_{1} + \mathbf{w}_{1} = \mathbf{v}_{2} + \mathbf{w}_{2} $ $ \implies $ $\mathbf{v}_{1} - \mathbf{v}_{2} = \mathbf{w}_{2} - \mathbf{w}_{1} \in W.$

$\Leftarrow$ $ \mathbf{v}_{1} - \mathbf{v}_{2} \in W \implies \mathbf{v}_{1} - \mathbf{v}_{2} = \mathbf{w}$ for some $\mathbf{w}$, $\mathbf{v}_{1} = \mathbf{v}_{2} + \mathbf{w}$ and $ \mathbf{v}_{2} = \mathbf{v}_{1} - \mathbf{w}$.

Let $ x \in \mathbf{v}_{1} + W_{1}$, then $ x = \mathbf{v}_{1} + \mathbf{w}_{1} = \mathbf{v}_{2} + (\mathbf{w} + \mathbf{w}_{1}) \in \mathbf{v}_{2} + W$.

Similarly, $\mathbf{v}_{2} + W \subseteq \mathbf{v}_{1} + W$.

Answer 1

Yes, it is correct. Note that there is a typo at the statement. You wrote “$\overline{\mathbf{v}}_i=\overline{\mathbf{v}}_i+W$”, but I guess that you meant “$\overline{\mathbf{v}}_i=\mathbf{v}_i+W$”.