A Question on Showing that a vector is a member of a coset

Question

Let $x,n_i,w$ be vectors in a vector space $X$ and $N$ be a subspace of $X$. $A_x = \{w \, | \, w = x + n_i, n_i \in N\}$ are the cosets of $N$. I want to show that if $v \in X$ then $v$ must be in one of the cosets. So I write: $v = v - n_1 + n_1$. If I let $y = v - n_1$ then $v \in A_y$. I'm worried though that this depended on the choice of $n_1$. Suppose I took $n_1 \neq n_2$ then $$v -n_1 \neq v- n_2 \implies y \neq v- n_2$$ So now for a different choice of $n_i$ I have that $v \notin A_y$. This doesn't seem correct. Shouldn't I have that $v$ is a member of the same coset, regardless of representative I choose for $n_i \in N$?

Answer 1

Since there's the equivalence relation $[a]=[b]\Rightarrow A_a=A_b (a,b\in V)$, hence $v-n_2\neq v-n_1=y$ doesn't lead to $v\not\in A_y$. As you can see, $A_{v-n_1}=A_{v-n_2}$, since $A_{v-n_1}\ni (v-n_1)+\underset{\in N}{\underbrace{n_0}}=(v-n_2)+\underset{\in N}{\underbrace{(n_2-n_1+n_0)}}\in A_{v-n_2} (\forall n_0\in N)$ and vice versa.