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→Generalizations
== Generalizations ==
== Generalizations ==
The above discussion only considers vector spaces in which the scalars are real numbers, but we could just as well talk about the set of polynomials with complex coefficients, where we multiple by complex scalars. More generally, given any [[field]] <math>k</math>, a vector space over <math>k</math> is an additive group in which addition is commutative and with which is associated a field of scalars, as the field of real numbers, such that the product of a scalar and an element of the group or a vector is defined, the product of two scalars times a vector is associative, one times a vector is the vector, and two distributive laws hold. In terms of another definition, a vector space is simply a module for which the ground ring is a field.
The above discussion only considers vector spaces in which the scalars are real numbers, but we could just as well talk about the set of polynomials with complex coefficients, where we multiple by complex scalars. More generally, given any [[field]] <math>F</math>, a vector space over <math>F</math> is an additive group in which addition is commutative and with which is associated a field of scalars, as the field of real numbers, such that the product of a scalar and an element of the group or a vector is defined, the product of two scalars times a vector is associative, one times a vector is the vector, and two distributive laws hold. In terms of another definition, a vector space is simply a module for which the ground ring is a field.
Specifically, let V be vector space over a field F. Then for all '''u''','''v''','''w''' ∈ V and a,b ∈ F the following axioms are obeyed:
'''Vector Addition'''
<br/>
1. [[Commutative property|Commutativity]]: '''u''' + '''v''' = '''v''' + '''u'''.
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2. [[Associative property|Associativity]]: ('''u''' + '''v''') + '''w''' = '''u''' + ('''v''' + '''w''').
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3. [[Identity_Element|Identity]]: There exists a '''0''' ∈ V such that '''v''' + '''0''' = '''0''' + '''v''' = '''v'''.
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4. [[Inverse]]: For all '''v''' there exists a (-'''v''') such that '''v''' + (-'''v''') =(-'''v''') + '''v''' = '''0'''.
'''Scalar Multiplication'''
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5. [[Associative property|Associativity]]: a(b'''v''') = (ab)'''v'''.
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6. [[Identity_Element|Identity]]: For 1 ∈ F (i.e, the multiplicative identity of F) it follows that 1'''v''' = '''v'''.
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[[Distributive property|'''Distributivity''']]
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7. Scalar sums: (a + b)'''v''' = a'''v''' + b'''v'''.
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8. Vector sums: a('''v''' + '''w''') = a'''v''' + a'''w'''.
==References==
Weisstein, Eric W. "Vector Space." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/VectorSpace.html
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Algebra]]
[[Category:Physics]]
[[Category:Physics]]