= Vector subspace
= Linear subspace
{synonym}
A vector subspace is a nonempty subset of a <vector space> closed under <vector addition> and <scalar multiplication>, and is therefore itself a <vector space>. Equivalently, whenever $u,v$ belong to the subset, so does $au+bv$ for all allowed <scalars> $a,b$. In particular it contains the <zero vector>.
For example, the plane $W=\{(x,y,z):x+y+z=0\}\subseteq\mathbb R^3$ is a <vector subspace>, with <basis> $(1,-1,0),(1,0,-1)$. Indeed every element is $(-y)(1,-1,0)+(-z)(1,0,-1)$. The parallel plane $x+y+z=1$ misses the <zero vector> and is an <affine subspace> instead. Homogeneous constraints produce <vector subspaces>: $W$ is the <kernel> of the <linear map> $(x,y,z)\mapsto x+y+z$. This lets us describe all solutions of a homogeneous <linear equation> through a <basis> of its <kernel>.
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