Let
be a field (or, more generally, a division ring). A vector space
over
is a set with two operations,
and
, such that
for all ![{\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} \in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7a7338b8ec9ae013ac86a006902e6f533d6d34)
for all ![{\displaystyle \mathbf {u} ,\mathbf {v} \in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55d35c94c8d55735e299fa26c8d298f8c7d9d107)
- There exists an element
such that
for all ![{\displaystyle \mathbf {u} \in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9760c4f376b6a157440323c062e96eb26f03d7b)
- For any
, there exists an element
such that ![{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7433ea1554bf647171daaf40cc8a70e8f3396ca4)
for all
and ![{\displaystyle \mathbf {u} \in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9760c4f376b6a157440323c062e96eb26f03d7b)
for all ![{\displaystyle \mathbf {u} \in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9760c4f376b6a157440323c062e96eb26f03d7b)
for all
and ![{\displaystyle \mathbf {u} ,\mathbf {v} \in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55d35c94c8d55735e299fa26c8d298f8c7d9d107)
for all
and ![{\displaystyle \mathbf {u} \in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9760c4f376b6a157440323c062e96eb26f03d7b)
Equivalently, a vector space is a module
over a ring
which is a field (or, more generally, a division ring).
The elements of
are called vectors, and the element
is called the zero vector of
.
This entry is a copy of the GNU FDL vector space article from PlanetMath. Author of the original article: djao. History page of the original is here