A vector space (over
) consists of a set
along with
two operations "
" and "
" subject to these conditions.
- For any
.
- For any
.
- For any
.
- There is a zero vector
such that
for all
.
- Each
has an additive inverse
such that
.
- If
is a scalar, that is, a member of
and
then the scalar multiple
is in
.
- If
and
then
.
- If
and
, then
.
- If
and
, then
.
- For any
,
.
Remark: Because it involves two kinds of addition and two kinds of multiplication, that definition may seem confused. For instance, in condition 7 "
", the first "
" is the real number addition operator while the "
" to the right of the equals sign represents vector addition in the structure
. These expressions aren't ambiguous because, e.g.,
and
are real numbers so "
" can only mean real number addition.