البوابة
من طرف 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) | |
 |  |
عدل سابقا من قبل Asmaa Mahmoud في الجمعة 25 يوليو 2008, 5:32 pm عدل 1 مرات (السبب : لتنسيق الموضوع)
