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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) = (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**, then*V* is a
**real vector space **.
If *, µ ***C**, then*V* 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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