vector space

موضوع: vector space الخميس يوليو 24, 2008 3:23 pm

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 :
1. Commutativity:

(1)

2. Associativity of vector addition:

(2)

3. Additive identity: For all ,

(3)

4. Existence of additive inverse: For any , there exists a such that

(4)

5. Associativity of scalar multiplication:

(5)

6. Distributivity of scalar sums:

(6)

7. Distributivity of vector sums:

(7)

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

(9)

and the number of distinct -dimensional subspaces of is

		(10)

		(11)

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واخر دعوانا ان الحمد لله رب العالمين

موضوع: رد: vector space السبت يوليو 26, 2008 12:08 pm

thank's alot shady
sorry but i have one qut.
this topic in geometry or topology
i think in top
then vector space the same of the meaning of top. space?????????
shady this properties remember me by function analysis right

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