Vector Spaces

Vector spaces. A vector space

is a set with two operations, addition and scalar multiplication, which satisfies the following axioms (or rules):

$(\mathbf{u}+\mathbf{v})+\mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w})$ for all $\mathbf{u},\mathbf{v},\mathbf{w} \in V$
$\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$ for all $\mathbf{u},\mathbf{v}\in V$
There exists a zero vector $\mathbf{0} \in V$ such that $\mathbf{u}+\mathbf{0}=\mathbf{u}$ for all $\mathbf{u} \in V$
For any $\mathbf{u} \in V$ , there exists an element $\mathbf{v} \in V$ such that $\mathbf{u}+\mathbf{v}=\mathbf{0}$
$a (b \mathbf{u}) = (a b) \mathbf{u}$ for all scalars and $\mathbf{u} \in V$
$1 \mathbf{u} = \mathbf{u}$ for all $\mathbf{u} \in V$
$a (\mathbf{u}+\mathbf{v}) = (a \mathbf{u}) + (a \mathbf{v})$ for all scalar and $\mathbf{u},\mathbf{v}\in V$
$(a+b) \mathbf{u} = (a \mathbf{u}) + (b \mathbf{u})$ for all scalars and $\mathbf{u} \in V$

Subspaces. If is a subset of and it is itself a vector space, then is said to be a vector subspace of . In particular if $\mathbf{u} + \mathbf{v},\, a \mathbf{u} \in W$ for any scalar and any $\mathbf{u},\mathbf{v} \in W$ then is a subspace of . In every vector space, $\{\mathbf{0}\}$ is a vector subspace, and is called the zero subspace. Let $\mathbf{v}_1,\ldots,\mathbf{v}_n \in V$ . Then the collection of all the linear combinations

$\displaystyle a_1\mathbf{v}_1+\dots+a_n\mathbf{v}_n$

with scalars

's becomes a subspace of

, denoted by

$\displaystyle \mathrm{span}\{\mathbf{v}_1,\ldots,\mathbf{v}_n\}.$