Page 5 - Vector Analysis

Chapter 1

Linear Algebra

1.1 Vector Spaces

Definition 1.1 (Vector spaces). A vector space V over a scalar field F is a set of elements
called vectors, together with two operations + : V ˆ V Ñ V and ¨ : F ˆ V Ñ V, called the
vector addition and scalar multiplication respectively, such that

1. v + w = w + v for all v, w P V.
2. (u + v) + w = u + (v + w) for all u, v, w P V.
3. There is a zero vector 0 such that v + 0 = v for all v P V.
4. For every v in V, there is a vector w such that v + w = 0.
5. α ¨ (v + w) = α ¨ v + α ¨ w for all α P F and v, w P V.
6. α ¨ (β ¨ v) = (αβ) ¨ v for all α, β P F and v P V.
7. (α + β) ¨ v = α ¨ v + β ¨ v for all α, β P F and v P V.
8. 1 ¨ v = v for all v P V.
For notational convenience, we often drop the ¨ and write αv instead of α ¨ v.
Remark 1.2. In property 4 of the definition above, it is easy to see that for each v, there
is only one vector w such that v + w = 0. We often denote this w by ´v, and the vector
substraction ´ : V ˆ V Ñ V is then defined (or understood) as v ´ w = v + (´w).

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