Subspaces

Let be a set of vectors in the vector space and inherit ‘s scalar multiplication and addition operations, and if and those operations themselves form a vector space, we say that is a subspace of .

The intersection of two subspace is another subspace.

Proof

Say we have two subspaces and for a vector space . Let and be two arbituary vectors in . The linear combination of and is still in because is a subspace. Similarly, the linear combination is also in . Thus, the linear combination of and is in , so we can say that is also a vector space and a subspace of .

Four often used fundamental subspaces are column space, row space, nullspace, and left null space.