Category:Quotient Vector Space is Vector Space

This category contains pages concerning Quotient Vector Space is Vector Space:

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.