# CHAPTER 5LINEAR ALGEBRA

Vectors are usually introduced with their geometric definition as directed line segments. Introduction of a coordinate system allows the concept of vector to be extended to a much broader class of objects called tensors, which are defined in terms of their transformation properties. As a tensor, vectors are now classified as first‐rank tensors. In dimensions, a given vector can be written as the linear combination of linearly independent basis vectors. Linear algebra is essentially the branch of mathematics that uses the concept of linear combination to extend the vector concept to a much broader class of objects. In this chapter, we discuss abstract vector spaces, which paves the way to many scientific and engineering applications of linear algebra.

## 5.1 FIELDS AND VECTOR SPACES

We start with the basic definitions used throughout this chapter [1–3]. As usual, a collection of objects is called a **set**. The set of all real numbers is denoted by and the set of all complex numbers by . The set of all ‐tuples of real numbers,

is shown by . Similarly, the ...

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