Quotient Vector Space

Last updated Nov 1, 2022

Let $X$ be a Vector Space over Field $F$ and let $Y \subset X$ be a Vector Subspace. Denote $X / Y$ to be the Equivalence Classes of the Equivalence Relation $\sim$ where $x \sim x$ if $x-x’ \in Y$ for $x,x’ \in X$. Then $X / Y$ is a Vector Space over $F$ with the following properties

1. $[0]$ is the Additive Identity of $X / Y$
2. For $ , [y] \in X/Y$, $ + [y] := [x+y]$
3. For $ \in X/Y$, $c \in F$, $c := [cx]$

TODO We need to show

• $\sim$ is an Equivalence Relation
• Vector Space operations are well defined
• Vector Space Axioms hold