Vector Analysis and Vector Space
The components of a vector need not be constants but can also be variables and functions of variables. For example, the position of a body moving through space can be described by a vector whose x, y, and z components are each functions of time. The methods of the calculus may be applied to such vector functions, leading to the branch of mathematics known as vector analysis.
The more general extension of vectors leads to the concept of a vector space. A vector space is a set of elements, A,B,C, … , called vectors, for which the operations of addition of vectors and multiplication of a vector by a scalar are defined and which satisfies ten axioms relating to such properties as closure under both operations, associativity, commutativity, and existence of a zero vector, an additive inverse (negative of a vector), and a unit scalar.
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