Subspaces, Affine sets, Convex sets, Cones

John Mitchell

- Let v
^{1},…,v^{k}be k vectors in IR^{n}. Let λ_{ 1},…,λ_{k}be k scalars. The vector v := ∑_{i=1}^{k}λ_{ i}v^{i}is a linear combination of v^{1},…,v^{k}. - Let S be a subset of IR
^{n}. S is a subspace if it is closed under linear combinations. Thus, for any k > 0, for any vectors v^{1},…,v^{k}S, and for any scalars λ_{ 1},…,λ_{k}, the linear combination v := ∑_{i=1}^{k}λ_{ i}v^{i}is also in S. Notice that the origin is in any nonempty subspace — just take all λ_{i}= 0. - The row space, range, and null space of a matrix are all subspaces.
- Let v
^{1},…,v^{k}be k vectors in IR^{n}. Let λ_{ 1},…,λ_{k}be k scalars satisfying ∑_{i=1}^{k}λ_{ i}= 1. (Note: some of the scalars may be negative.) The vector v := ∑_{i=1}^{k}λ_{ i}v^{i}is an affine combination of v^{1},…,v^{k}. - Let S be a subset of IR
^{n}. S is an affine space if it is closed under affine combinations. Thus, for any k > 0, for any vectors v^{1},…,v^{k}S, and for any scalars λ_{ 1},…,λ_{k}satisfying ∑_{i=1}^{k}λ_{ i}= 1, the affine combination v := ∑_{i=1}^{k}λ_{ i}v^{i}is also in S. - The set of solutions to the system of equations Ax = b is an affine space.
- An affine space is a translation of a subspace.
- Any subspace is also an affine space.
- Let v
^{1},…,v^{k}be k vectors in IR^{n}. Let λ_{ 1},…,λ_{k}be k nonnegative scalars satisfying ∑_{i=1}^{k}λ_{ i}= 1. The vector v := ∑_{i=1}^{k}λ_{ i}v^{i}is a convex combination of v^{1},…,v^{k}. - Let S be a subset of IR
^{n}. S is an convex set if it is closed under convex combinations. Thus, for any k > 0, for any vectors v^{1},…,v^{k}S, and for any nonnegative scalars λ_{ 1},…,λ_{k}satisfying ∑_{i=1}^{k}λ_{ i}= 1, the convex combination v := ∑_{i=1}^{k}λ_{ i}v^{i}is also in S. - A hyperplane is the set of solutions to a linear equality a
^{T }x = b with a IR^{n}\{0}. - A half-space is the set of solutions to a linear inequality a
^{T }x ≥ b with a IR^{n}\{0}. - A polyhedron is the intersection of a finite number of half-spaces. 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 IR
^{n}. S is a cone if it is closed under nonnegative scalar multiplication. Thus, for any vector v S and for any nonnegative scalar λ, the vector λv is also in S. - Let S be a subset of IR
^{n}. 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 v^{1},…,v^{k}S, and for any nonnegative scalars λ_{ 1},…,λ_{k}, the linear combination v := ∑_{i=1}^{k}λ_{ i}v^{i}is also in S. - The origin is in any nonempty cone — just take λ = 0.
- Any subspace is a convex cone.