Title:

Author:

Linear Algebra

Copyright:

T. Tsujioka

Advanced Seminar

**Linear Algebra**

Chapter 1. Vectors

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

Algebraically, ** vectors** are defined as ordered

Notations for vectors vary in textbooks; in bold face:

under bar:

can easily extend the concept to

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** = (

= (a+ba_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3}).

Let any scalar be *k*, then the scalar multiplication is defined by *k a* =(