A vector space is a set of elements, called vectors, that can be added together and multiplied by scalars. A set of elements is a vector space if it satisfies the following axioms:
- Closure under addition: For any two vectors u and v in V, their sum u + v is also in V.
- Associativity of addition: For any three vectors u, v, and w in V, the following equation holds: (u + v) + w = u + (v + w).
- Commutativity of addition: For any two vectors u and v in V, the following equation holds: u + v = v + u.
- Existence of a zero vector: There exists a unique vector 0 in V such that for any vector u in V, the following equation holds: u + 0 = u.
- Additive inverse: For any vector u in V, there exists a unique vector -u in V such that the following equation holds: u + (-u) = 0.
- Closure under scalar multiplication: For any vector u in V and any scalar c, the product cu is also in V.
- Associativity of scalar multiplication: For any vector u in V and any two scalars c and d, the following equation holds: (cu)d = c(ud).
- Distributivity of scalar multiplication over vector addition: For any vector u and v in V and any scalar c, the following equation holds: c(u + v) = cu + cv.
- Distributivity of scalar multiplication over scalar addition: For any vector u in V and any two scalars c and d, the following equation holds: (c + d)u = cu + du.
- Identity element for scalar multiplication: For any vector u in V, the following equation holds: 1u = u.
Vector spaces are used in many areas of mathematics, including linear algebra, geometry, and analysis. They are also used in many applications in physics, engineering, and computer science.Here are some of the benefits of using vector spaces:
