Quotient Vector Space is Vector Space

Theorem
Let $K$ be a field.

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

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

Define:


 * $X/N = \set {x + N : x \in X}$

where $x + N$ is the Minkowski sum of $x$ and $N$.

Define:


 * $\paren {x + N} +_{X/N} \paren {y + N} = \paren {x + y} + N$

for $x, y \in X$, and:


 * $\alpha \circ_{X/N} {x + N} = \paren {\alpha x} + N$

for $\alpha \in K$ and $x \in X$.

Then $\paren {X/N, +_{X/N}, \circ_{X/N} }_K$ is a vector space.

Vector Addition is Well-Defined
To show that $+_{X/N}$ is well-defined, we show that if $x, x', y, y' \in N$ are such that:


 * $x + N = x' + N$

and:


 * $y + N = y' + N$

we have:


 * $\paren {x + y} + N = \paren {x' + y'} + N$

Since $x + N = x' + N$, we have:


 * $x' \in x + N$

and so there exists $z \in N$ such that $x + z = x'$.

Then $z = x' - x$, and so $x' - x \in N$.

Similarly, we have that $y' - y \in N$.

So we have $\paren {x' + y'} - \paren {x + y} \in N$, since $N$ is a linear subspace of $X$, so that:


 * $N = \paren {x' + y'} - \paren {x + y} + N$

by the lemma.

So that:


 * $\paren {x + y} + N = \paren {x' + y'} + N$

Scalar Multiplication is Well-Defined
Let $\alpha \in K$.

Suppose that $x, x' \in N$ are such that:


 * $x + N = x' + N$

We need to show that:


 * $\alpha x + N = \alpha x' + N$

As argued in proving vector addition is well-defined, we have that $x' - x \in N$.

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


 * $\alpha \paren {x' - x} = \alpha x' - \alpha x \in N$

So from the lemma, we have:


 * $N = \alpha x' - \alpha x + N$

So that:


 * $\alpha x + N = \alpha x' + N$

Proof of
Let $\mathbf x, \mathbf y \in X/N$.

Then there exists $x, y \in N$ such that $\mathbf x = x + N$ and $\mathbf y = y + N$.

Then we have:

Proof of
Let $\mathbf x, \mathbf y, \mathbf z \in X/N$.

Then there exists $x, y, z \in N$ such that $\mathbf x = x + N$, $\mathbf y = y + N$ and $\mathbf z = z + N$.

Then we have:

Proof of
Let $0_{X/N} = 0_X + N = N$.

We prove that $0_{X/N}$ satisfies the demand of.

Let $\mathbf x \in X/N$.

Then there exists $x \in X$ such that $\mathbf x = x + N$.

We have:

Proof of
Let $\mathbf x \in X/N$.

Then there exists $x \in X$ such that $\mathbf x = x + N$.

Set $-\mathbf x = -x + N$.

We then have:

Proof of
Let $\lambda, \mu \in K$.

Let $\mathbf x \in X/N$.

Then there exists $x \in X$ such that $\mathbf x = x + N$.

Then we have:

Proof of
Let $\lambda \in K$.

Let $\mathbf x, \mathbf y \in X/N$.

Then there exists $x, y \in N$ such that $\mathbf x = x + N$ and $\mathbf y = y + N$.

We then have:

Proof of
Let $\lambda, \mu \in K$.

Let $\mathbf x \in X/N$.

Then there exists $x \in X$ such that $\mathbf x = x + N$.

We have:

Proof of
Let $\mathbf x \in X/N$.

Then there exists $x \in X$ such that $\mathbf x = x + N$.

Then we have: