Subspaceis a subset of the vector space, which is also called a linear subspace or vector subspace.

Subspace Decision Theorem

Let V be a vector space over field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following three conditions:

• The zero vector is in W;
• If u and v are elements of W, then the vector sum u + v is an element of W;
• If u is an element of W and c is a scalar of K, then the scalar product cu is an element of W.

Properties of subspace

• For every vector space V, the set { 0 } and V itself are subspaces of V;
• If V is an inner product space, then the orthogonal complement of any subspace of V is also a subspace;
• The intersection of any number of vector subspaces is still a vector subspace;
• One way to characterize subspaces is that they are closed under linear combinations.

Related words: vector space