There are two algebraic operations done with elements of Fn. One is addition and the other is multiplication by numbers, called scalars. In the case of ℂn the scalars are complex numbers while in the case of ℝn the only allowed scalars are real numbers. Thus, the scalars always come from F in either case.
This is known as scalar multiplication. If x,y ∈ Fn then x + y ∈ Fn and is defined by
|x + y|| =
| ≡ ||(1.10)|
With this definition, the algebraic properties satisfy the conclusions of the following theorem.
the commutative law of addition,
the associative law for addition,
the existence of an additive identity,
the existence of an additive inverse, Also
In the above 0 = (0,
You should verify that these properties all hold. As usual subtraction is defined as x − y ≡ x+