Linear Algebra

Chapter 1. Vectors

1-1. Vectors and Scalars: Definition of Vectors --- Algebraic Vectors and Geometrical Vectors

Algebraically, vectors are defined as ordered n-tuples of numbers, such as (2, 3, -1, 7, ..., 5), (a₁, a₂, a₃, ..., a_n).
Notations for vectors vary in textbooks; in bold face: a, with an arrow: , with an angled bracket:, with an
under bar: a. For the sake of simplicity, we restrict ourselves only on 2- or 3-tuple in the following sections. You
can easily extend the concept to n-dimensions.

Fig. 1-1

Alternative definition is to express vectors as geometrical arrows as in Fig.1-1. Historically, the concept of vectors
arose from the necessity of treating velocity or forces on mathematically rigorous foundations. Unlike just numbers, those concepts have two properties: the magnitude and the direction. Grasping operations on vectors as a mapping into a different set of vectors (---called a vector space), an extensive applicability has been found and organized into linear algebra ---one of the basic subjects in mathematics. If you acquire the knowledge of linear algebra, you will understand why some simultaneous linear equations have infinite solutions or no solution, how 3-D objects such as straight lines, planes, and spheres can be expressed simply, and how some functions can be expanded by use of orthogonal functions, to name a few. Furthermore, you will learn that calculus in 3-D, or any dimension, can be
treated simply by combining vector operators. We need these vector operations to treat objects in real world. After all, the real world objects we observe exist intrinsically in multi-dimensions, not just 1-dimension.

In order to distinguish numbers from vectors, ordinary numbers are called scalars.

1-2. Vector Arithmetic Rules: Vector Algebra --- Addition and Scalar Multiplication

Given a = (a₁, a₂, a₃), b = (b₁, b₂, b₃), the addition of vectors is defined algebraically as

a + b = (a₁+ b₁, a₂+ b₂, a₃+ b₃).

Let any scalar be k, then the scalar multiplication is defined by ka =(ka₁, ka₂, ka₃). When these two operations are defined, it is shown that vectors always obey a set of rules named as the axiom of vector space. You are accustomed to manipulating numbers with addition and subtraction. If you treat each component as a number, and manipulate vectors each component by component, you will soon realize that there is no difference between the manipulation of vectors and the manipulation of numbers. In this sense, vectors may be considered multi-dimensional numbers.