This document describes various subsets of .

- Let
be
*k*vectors in . Let be*k*scalars. The vector is a**linear combination**of . - Let
*S*be a subset of .*S*is a**subspace**if it is closed under linear combinations. Thus, for any*k*>0, for any vectors , and for any scalars , the linear combination is also in*S*. Notice that the origin is in any nonempty subspace -- just take all . - The row space, range, and null space of a matrix are all subspaces.
- Let
be
*k*vectors in . Let be*k*scalars*satisfying*. (Note: some of the scalars may be negative.) The vector is an**affine combination**of . - Let
*S*be a subset of .*S*is an**affine space**if it is closed under affine combinations. Thus, for any*k*>0, for any vectors , and for any scalars satisfying , the affine combination is also in*S*. - The set of solutions to the system of equations
*Ax*=*b*is an affine space. This is why we talk about affine spaces in this course! - An affine space is a translation of a subspace.
- Any subspace is also an affine space.
- Let
be
*k*vectors in . Let be*k**nonnegative*scalars satisfying . The vector is a**convex combination**of . - Let
*S*be a subset of .*S*is an**convex set**if it is closed under convex combinations. Thus, for any*k*>0, for any vectors , and for any nonnegative scalars satisfying , the convex combination is also in*S*. - Polyhedra are convex sets.
- A polytope is defined to be a bounded polyhedron. Note that every point in a polytope is a convex combination of the extreme points.
- Any subspace is a convex set. Any affine space is a convex set.
- Let
*S*be a subset of .*S*is a**cone**if it is closed under nonnegative scalar multiplication. Thus, for any vector and for any nonnegative scalar , the vector is also in*S*. - Let
*S*be a subset of .*S*is a**convex cone**if it is a cone and it is convex. It can be shown that this is equivalent to saying that*S*is closed under nonnegative linear combinations. Thus, for any*k*>0, for any vectors , and for any nonnegative scalars , the linear combination is also in*S*. - The origin is in any nonempty cone -- just take .
- Any subspace is a convex cone.

John E. Mitchell