A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is -dimensional Euclidean space , where every element is represented by a list of real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.
For a general vector space, the scalars are members of a field , in which case is called a vector space over .
Euclidean -space is called a real vector space, and is called a complex vector space.
In order for to be a vector space, the following conditions must hold for all elements and any scalars :
2. Associativity of vector addition:
3. Additive identity: For all ,
4. Existence of additive inverse: For any , there exists a such that
5. Associativity of scalar multiplication:
6. Distributivity of scalar sums:
7. Distributivity of vector sums:
8. Scalar multiplication identity:
Let be a vector space of dimension over the field of elements (where is necessarily a power of a prime number). Then the number of distinct nonsingular linear operators on is
and the number of distinct -dimensional subspaces of is
عدل سابقا من قبل Asmaa Mahmoud في الجمعة 25 يوليو 2008, 5:32 pm عدل 1 مرات (السبب : لتنسيق الموضوع)