Source: https://nostalgia.wikipedia.org/wiki/Vector_Space
Timestamp: 2019-04-25 06:20:25+00:00

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The fundamental concept in linear algebra is that of a vector space or linear space. It is a generalization of the set of all geometrical vectors and is used throughout modern mathematics.
V is closed under vector addition.
Associativity of vector addition in V.
There exists an element 0 in V, such that for all elements v in V, v+0=v.
Existence of an additive identity element in V.
For all v in V, there exists an element -v in V, such that v+(-v)=0.
Existence of additive inverses in V.
Commutativity of vector addition in V.
V is closed under scalar multiplication.
Associativity of scalar multiplication in V.
If 1 denotes the multiplicative identity of the field F, then 1*v=v.
Distributivity with respect to vector addition.
Distributivity with respect to field addition.
for all a in F and v in V.
The members of a vector space are called vectors. The concept of a vector space is entirely abstract like the concepts of a group, ring, and field. To determine if a set V is a vector space one must specify the set V, a field F and define vector addition and scalar multiplication in V. Then if V satisfies the above 10 properties it is a vector space over the field F.
A vector space over R, the set of real numbers, is called a real vector space.
A vector space over C, the set of complex numbers, is called a complex vector space.
Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is called linearly independent. A linearly independent set whose span is the whole space is called a basis.
All bases for a given vector space have the same cardinality. Using Zorn's Lemma, it can be proved that every vector space has a basis, and vector spaces over a given field are fixed up to isomorphism by a single cardinal number (called the dimension of the vector space) representing the size of the basis. For instance the real vector spaces are just R0, R1, R2, R3, ..., R∞, ... As you would expect, the dimension of the real vector space R3 is three.
A morphism from a vector space V to a vector space W (necessarily over the same field) is called a linear transformation or "linear map". That is, a map is linear if and only if it preserves sums and scalar products. An isomorphism is a linear map that is one-to-one and onto. The set of all linear maps from V to W is denoted L(V,W) and makes up a vector space over the same field. When bases for both V and W are given, linear maps can be expressed in terms of components as matrices.
In abstract algebra, the concept of a vector space is generalized to modules by replacing the underlying field F by a commutative ring and retaining the above 10 axioms.

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