Quotient vector space and codimension

A vector space of cosets modulo a subspace; its dimension defines codimension

Quotient vector space and codimension

Let $X$ be a vector space over $K$ , and let $Y\subset X$ be a linear subspace .

Define a relation on $X$ by

x\sim u \quad\Longleftrightarrow\quad x-u\in Y.

This is an equivalence relation. The equivalence class of $x$ is denoted

[x]=x+Y:=\{u\in X\mid u-x\in Y\}.

The set of all equivalence classes is written

X/Y:=\{x+Y\mid x\in X\}.

Define operations on $X/Y$ by

(x+Y)+(u+Y):=(x+u)+Y,\qquad \lambda(x+Y):=(\lambda x)+Y.

These operations are well-defined (independent of representatives) and make $X/Y$ a vector space, called the quotient vector space.

The codimension of $Y$ in $X$ is defined by

\operatorname{codim}(Y):=\dim(X/Y).

Examples:

If $Y=\{0\}$ , then $X/Y\cong X$ via $x+\{0\}\mapsto x$ .
If $X=\mathbb{R}^2$ and $Y$ is the $x$ -axis, then $X/Y$ is (linearly) isomorphic to $\mathbb{R}$ .
If $Y=X$ , then $X/Y$ is the zero vector space.