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2. Vector space

2.1. Definitions and examples

A vector space is a set V of objects, called vectors, on which are defined two operations, called addition and scalar multiplication, such that the following conditions are satisfied:

1) + = + ,   , V
2) ( + ) + = + ( + ),   , , V
3) unique V;   + = , V
4) for every V   ' V such that + ' =
5) ) = (µ),    V ja , µ R
6) ( + µ) = + µ,    V ja , µ R
7) ( + ) = + ,    V ja R
8) 1 = ,    V.

If the above numbers , µ R, thenV is a real vector space . If , µ C, thenV is a complex vector space . In the sequel a vector space means a real or a comples vector space.

Example 1: A real and a complex vector space
Example 2: The set of m x n matrices form a vector space.


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