Prove that $v_{1} + W = v_{2} + W$ if and only if $v_{1} - v_{2}\in W$.

Question

Let $W$ be a subspace of a vector space $V$ over a field $\textbf{F}$. For any $v\in V$ the set $\{v\} + W = \{v+w:w\in W\}$ is called the coset of $W$ containing $v$. It is customary to denote this coset by $v + W$ rather than $\{v\} + W$.

(a) Prove that $v + W$ is a subspace of $V$ if and only if $v\in W$.

(b) Prove that $v_{1} + W = v_{2} + W$ if and only if $v_{1} - v_{2}\in W$.

MY ATTEMPT

(a) If $v\in W$, then $v + W = W$, which is a subspace by assumption.

If $v + W$ is a subspace of $V$, then $0\in v + W$.

Thus there is a $w\in W$ such that $v + w = 0$, that is to say, $v = -w\in W$, and we are done.

(b) If $v_{1} + W = v_{2} + W$, then $v_{1}\in v_{2} + W$.

Consequently, $v_{1} = v_{2} + w$, that is to say, $v_{1} - v_{2} = w \in W$, and we are done.

Conversely, if $v_{1} - v_{2}\in W$, then $v_{1} - v_{2} = w\in W$. Hence $v_{1} = v_{2} + w$.

Then we have that $v_{1} + W = (v_{2} + w) + W = v_{2} + W$.

Any comments on my sotution? Any contribution is appreciated.

Answer 1

You can think this geometrically. For example, in $\mathbb{R}^2$. The only subspaces are $0, \mathbb{R}^2$ and Lines passing through origin. Take a non- trivial subspace $W$. Then, $W$ is a line passing through origin. Now, $v+W$ is a translate of $W$. So, if $v \notin W$ then we have translated $W$ away from origin $\implies W$ cannot contain the zero vector, hence, fails to be a subspace of $\mathbb{R}^2$.

Answer 2

Notice that you can restate your question in terms of equivalence relations. Let $R$ be the relation defined on $V$ by, for all $v_1,v_2\in V, v_1Rv_2$ if and only if $v_1-v_2\in W$. This way, you second statement becomes the obvious fact that two vectors $v_1$ and $v_2$ are related under $R$ if and only if they represent the same equivalence class.