5.2. ALGEBRA IN Rn 55

5.2 Algebra in Rn

There are two algebraic operations done with points of Rn. One is addition and the other ismultiplication by numbers, called scalars. Yes, numbers =scalars.

Definition 5.2.1 If x ∈Rn and a is a number, also called a scalar, then ax ∈Rn is definedby

ax= a(x1, · · · ,xn)≡ (ax1, · · · ,axn) . (5.1)

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

x+y = (x1, · · · ,xn)+(y1, · · · ,yn)

≡ (x1 + y1, · · · ,xn + yn) (5.2)

An element of Rn x ≡ (x1, · · · ,xn) is often called a vector. The above definition is knownas vector addition.

With this definition, the algebraic properties satisfy the conclusions of the followingtheorem. The conclusions of this theorem are called the vector space axioms. There aremany other examples.

Theorem 5.2.2 For v,w vectors in Rn and α,β scalars, (real numbers), the followinghold.

v+w=w+v, (5.3)

the commutative law of addition,

(v+w)+z = v+(w+z) , (5.4)

the associative law for addition,v+0= v, (5.5)

the existence of an additive identity

v+(−v) = 0, (5.6)

the existence of an additive inverse, Also

α (v+w) = α v+αw, (5.7)

(α +β ) v = α v+βv, (5.8)

α (βv) = αβ (v) , (5.9)

1v = v. (5.10)

In the above 0= (0, · · · ,0).

You should verify these properties all hold. For example, consider 5.7.

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) .