Quotient vector space ($V/W$)

For a vector subspace $W\le V$, the quotient vector space $V/W$ consists of cosets $v+W$, with vector operations induced from $V$.

The quotient map of vector spaces $\pi :V\to V/U$ sends a vector $v$ to its coset $v+U$, where $U$ is a linear subspace. It is a surjective linear map with kernel $U$: two vectors have the same image exactly when their difference lies in $U$.

For $V={\mathbb{R}}^2$ and $U=\mathrm{span}(1,0)$, the quotient retains only the second coordinate, so every horizontal line represents one quotient vector. More generally, a linear map $f:V\to W$ vanishing on $U$ factors uniquely as $f=\overline{f}\circ \pi$. The formula $\overline{f}(v+U)=f(v)$ is well defined because changing the representative by an element of $U$ leaves its value unchanged.

For a linear map $T:V\to V$ and an invariant subspace $U$, the rule $\overline{T}(v+U)=T(v)+U$ defines a linear map on the quotient vector space $V/U$. Indeed, two representatives differ by an element of $U$, so their images also differ by an element of $U$. Conversely, this rule is well-defined only if $T(U)\subseteq U$, by comparing every $u\in U$ with zero.

For a vector subspace $W$ of $X$, its codimension is the dimension of the quotient vector space $X/W$. This definition also applies when $X$ is infinite-dimensional. If $X=W\oplus E$ is a direct sum, the map $e\mapsto e+W$ is a linear isomorphism from $E$ to $X/W$, so ${\mathrm{codim}}_{X}W=\dim E$. In particular, the kernel of a nonzero linear functional has codimension one: normalize $Te=1$ and write $x=(x-(Tx)e)+(Tx)e$.

For fixed $u\ne 0$, the parallel affine lines $l_x=x+\mathrm{span}(u)$ are precisely the cosets of the indicated quotient vector space. Changing representatives adds a multiple of $u$, so $l_x+l_y=l_{x+y}$ and $a{l}_x=l_{ax}$ are well-defined. The zero is the whole line $l_0$, not a single point. If $(u,b_1,b_2)$ is a basis, the two line classes $l_{b_1},l_{b_2}$ form a basis of the quotient by spanning and independence modulo $\mathrm{span}(u)$.