Quotient Vector Space is Vector Space/Lemma
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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:
- $N = z + N$
$\blacksquare$