2.2. GEOMETRIC MEANING OF VECTORS 17

This is known as scalar multiplication. If x,y ∈ Fn then x+y ∈ Fn and is defined by

x+y = (x1, · · · ,xn)+(y1, · · · ,yn)

≡ (x1 + y1, · · · ,xn + yn) (2.2)

With this definition, vector addition and scalar multiplication satisfy the conclusionsof the following theorem. More generally, these properties are called the vector spaceaxioms.

Theorem 2.1.2 For v,w ∈ Fn and α,β scalars, (real numbers), the following hold.

v+w = w+v, (2.3)

the commutative law of addition,

(v+w)+ z = v+(w+ z) , (2.4)

the associative law for addition,v+0 = v, (2.5)

the existence of an additive identity,

v+(−v) = 0, (2.6)

the existence of an additive inverse, Also

α (v+w) = αv+αw, (2.7)

(α +β )v =αv+βv, (2.8)

α (βv) = αβ (v) , (2.9)

1v = v. (2.10)

In the above 0 = (0, · · · ,0).

You should verify these properties all hold. For example, consider 2.7

α (v+w) = α (v1 +w1, · · · ,vn +wn)

= (α (v1 +w1) , · · · ,α (vn +wn)) = (αv1 +αw1, · · · ,αvn +αwn)

= (αv1, · · · ,αvn)+(αw1, · · · ,αwn) = αv+αw.

As usual, subtraction is defined as x−y≡ x+(−y) .

2.2 Geometric Meaning Of VectorsThe geometric meaning is especially significant in the case of Rn for n = 2,3. Here is ashort discussion of this topic.