Page 802 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

form a basis. It is called the standard basis for . Since there are n vectors in this basis, is an n-dimensional vector
space.

Remark Do not confuse the complex number with the vector from the standard basis for (see

Example 3, Section 3.4). The complex number i will always be set in lightface type and the vector i in boldface.

EXAMPLE 2 Complex

In Example 3 of Section 5.1 we defined the vector space of matrices with real entries. The complex analog of this

space is the vector space of matrices with complex entries and the operations of matrix addition and scalar

multiplication. We refer to this space as complex .

EXAMPLE 3 Complex-Valued Function of a Real Variable
If and are real-valued functions of the real variable x, then the expression

is called a complex-valued function of the real variable x. Two examples are

(1)

Let V be the set of all complex-valued functions that are defined on the entire line. If and

are two such functions and k is any complex number, then we define the sum function and the

scalar multiple by

For example, if and are the functions in 1, then

It can be shown that V together with the stated operations is a complex vector space. It is the complex analog of the vector

space of real-valued functions discussed in Example 4 of Section 5.1.

EXAMPLE 4 Complex

Calculus Required is a complex-valued function of the real variable x, then f is said to be continuous if and

If