vector spaces

Vector Space over a Field

A vector space over a field $F$ is a non-empty set

V

with a binary operation

\oplus

on

V

and a binary function

\otimes : F \times V \to V

such that the following hold for any

a, b \in F

and

u, v \in V

$\oplus$ is associative and commutative
There is a identity element $0_{V}$ with respect to $\oplus$
$1_{F}$ is an identity element for $\otimes$
$\oplus$ has inverses for $0_{V}$
$\otimes$ distributes into $\oplus$
$(a +_{F} b) \otimes v = (a \otimes v) \oplus (b \otimes v)$
$a \otimes (b \otimes v) = (a \cdot_{F} b) \otimes v$

Subspace of a Vector Space

A subset

W

of a vector space

V

F

is called a subspace of

V

if

W

itself is a vector space over

F

with the same operations of vector addition and scalar multiplication as

V

. Specifically,

W

is a subspace if:

$0_{V} \in W .$
$\forall u, v \in W, u + v \in W$
$\forall u \in W, c \in F, c \cdot u \in W$