**Definition:** A __vector space__ is a nonempty set of objects, called __vectors__, on which are defined two operations, called additon and multiplication by scalars (real numbers), subject to the ten axioms (or rules) listed below. The axioms must holr for all vectors **u**, **v**, and **w** in V and for all scalars c and d.

1. The sum of **u** and **v**, denoted **u**+**v**, is in V.

2. **u**+**v**=**v**+**u**

3. (**u**+**v**)+**w**=**u**+(**v**+**w**)

4. There is a __zero__ vector **0** in V such that **u**+**0**=**u** ***0** is unique.

5. For each **u** in V, there is a vector -**u** in V such that **u**+(-**u**)=**0**. -**u** is unique and is called the negative of **u**.

6. The scalar multiple of **u** by c, denoted c**u**, is in V.

7. c(**u**+**v**)=c**u**+c**v**.

8. (c+d)**u**=c**u**+d**u**.

9. c(d**u**)=(cd)**u**.

10. 1⋅**u**=**u**

*for each **u** in V and scalar c:

- 0
**u**=**0** - c
**0**=**0** - -
**u**=(-1)**u**

**Example 1: ** The spaces **R**², where n≥1, are the premier examples of vector spaces. The geometric intuiton developed for **R**³ will help you understand and visualize many concepts throught the chapter.

**Example 2: **Let V be the set of all arrows(directed line segments) in three dimensional space, with two arrows regarded as equal if they have the same length and point in the same direction. Define additon by parallelogram rule and for each **v** in V, define c**v** to be the same direction. Define additon by

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