Within spaces ( ), we can define subspaces.

To give na example, a subspace of is a set of two-dimensional vectors within , where the set meets three specific conditions:

  1. The set includes the zero vector
  2. The set is closed under scalar multiplication
  3. The set is closed under addition

Keep in mind that the first condition, that a subspace must include the zero vector, is logically already included as part of the second condition, that a subspace is closed under multiplication.

A span is always a subspace.

Тhe zero vector is always a subspace.

Аn entire space is always a subspace of itself.