Vector Subspaces

Definition

Suppose $V$ is a subspace over $\F $. A vector space $U$ over $\F $ is a subspace (or linear subspace, vector subspace) of $V$ if $U \subset V$ and vector addition and scalar multiplication defined for $U$ agree with those defined for $V$. In other words, a subspace is a subset of a vector space which is closed under vector addition and scalar multiplication.

For example, the entire set of vectors is a subspace. As a second example, the set consisting only of the zero vector is a subspace; we call this the zero subspace. These two subspaces are the trivial subspaces. A nontrivial subspace is a subspace that is not trivial.

Notation

Let $(V, \F )$ be a vector space. Let $U \subset V$ with

\[ \alpha u + \beta v \in U \]

for all $\alpha , \beta \in \F $ and $u, v \in U$. Then $U$ is a subspace of $(V, \F )$.

Characterization

Suppose $V$ is a vector space over a field $\F $ and $U \subset V$. $U$ is a subspace if and only if $U$ satisifes

$0 \in U$ (contains additive identity)
$u + w \in U$ for all $u, w \in U$ (closed under addition)
$\alpha u \in U$ for all $\alpha \in \F $ and $u \in U$ (closed under scalar addition)

Properties

The intersection of a family of subspaces is a subspace.

There exists a family of subspaces whose union is not a subspace;

In other words: the union of a family subspaces need not be a subspace.

A subspace must contain the zero vector; in other words, every subspace is nonempty.