Euclidean Subspace

A Euclidean subspace is a subset S of Rn with the following properties:

1. The zero vector 0 is an element of S.
2. If u and v are elements of S, then u + v is an element of S.
3. If v is an element of S and c is a scalar, then cv is an element of S.

There are several common variations on these requirements, all of which are logically equivalent to the list above.

Because subspaces are closed under both addition and scalar multiplication, any linear combination of vectors from a subspace is again in the subspace. That is, if v₁, v₂, ..., v_k are elements of a subspace S, and c₁, c₂, ..., c_k are scalars, then

c₁ v₁ + c₂ v₂ + · · · + c_k v_k

is again an element of S.