Subspaces of Vector Spaces

Subspace

A nonempty subset W of a vector space V is called a subspace of V if W is itself a vector space under the operations of vector addition and scalar multiplication defined in V.

Nonempty

Contains at least one thing. Every valid vector space / subspace must contain zero vector.

Theorem 1 (Test for a Subspace)

Let W be a nonempty subset of a vector space V. Then W is a subspace of V if and only if the following two conditions hold:

(i) If u,v ∈ W, then u + v ∈ W

(ii) If u ∈ W and c is any scalar, then cu ∈ W

Check if Subset of Vector Space is Subspace

Zero Subspace

Since every vector space contains the zero vector, the set W = {0} is always a subspace of V. This subspace is called the zero subspace.

Trivial Subspace

The entire vector space V is also a subspace of itself. Thus, every vector space has at least the two subspaces {0} and V. These are sometimes called the trivial subspaces.

Nontrivial Subspace

A subspace W satisfying:

{0} “proper subset symbol” W “proper subset symbol” V

is called a proper subspace or a nontrivial subspace of V.

Subspace Order of Functions

Theorem 2 (The Intersection of Two Subspaces Is a Subspace)

If V and W are both subspaces of a vector space U, then V ∩ W is also a subspace of U.