Quotient Vector Space is Vector Space/Lemma

Lemma

Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $N$ be a linear subspace of $X$.

Let $z \in N$.

Then:

$z + N = N$

Proof

Since $N$ is a linear subspace, we have:

$z + N \subseteq N$

Conversely, let $x \in N$.

Then since $z \in N$ and $N$ is a linear subspace, we have $x - z \in N$.

Then:

$x = z + \paren {x - z} \in z + N$

So that:

$N \subseteq z + N$

giving:

$N = z + N$

$\blacksquare$