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Show that $\textbf{v} + W \subseteq W \Rightarrow W \subseteq

\textbf{v} + W.$

Here is my proof, is it correct? Is there any easier way?

Note that for any element $\mathbf{v + w}$ in $v+W$, $\mathbf{v+w}$ is in $W$ as well. Hence $\mathbf{v} + \mathbf{w} = \mathbf{w}_{1}$ for some $\mathbf{w_1} \in W$ Which implies $\mathbf{v} = \mathbf{w}_{1} -\mathbf{w} \in W$

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Now the proper proof starts :

Let $w_2 \in W$, now $ \mathbf{w}_{2} – \mathbf{v} = \mathbf{w}_{2} – (\mathbf{w}_{1}-\mathbf{w}) = \mathbf{w}_{3}$ for some $\mathbf{w}_{3}$. Hence $\mathbf{w}_{2} = \mathbf{v} + \mathbf{w}_{3} \in \mathbf{v} + W$. Thus $ W \subseteq \mathbf{v} + W$.

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Let $w \in W$. You have to show that $w \in v+W$.

My proof goes as follows:

$w \in W \Rightarrow -w \in W \Rightarrow v-w \in v+W \Rightarrow v-w \in W \Rightarrow w-v \in W \Rightarrow w \in v+W$.

I don’t understand your proof. You start with “Let $y\in\mathbf{v}+W$”, but what you should be proving is that $W\subset\mathbf{v}+W$, assuming that $\mathbf{v}+W\subset W$. Therefore, the proof should start with “Let $y\in W$”.

I would prove it as follows: since $\mathbf{v}+W\subset W$, $\mathbf{v}\in W$, because $\mathbf{v}=\mathbf{v}+0\in\mathbf{v}+W\subset W$. But then, since $W$ is a subspace, $\mathbf{v}+W\subset W$.

If you want to prove the reverse implication, it’s similar. If $W\subset\mathbf{v}+W$, then $0=\mathbf{v}+\mathbf{w}$, for some $\mathbf{v}\in W$. Therefore, $\mathbf{v}=-\mathbf{w}\in W$, which implies that $\mathbf{v}+W\subset W$.

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