## Linear Algebraic Geometry

Chapter 1. Determinants and Matrices — Overview

§1-1. Vector Space: the Set of Vector Objectsin *n*-D **R**^{n}(or **C**^{n})

*Vectors* are mathematical objects, each having properties of magnitude and direction. A number is called a *scalar* in order to distinguish from a vector. That is, scalars are mathematical objects, each having a magnitude only.

Vectors are a kind of *n*-dimensional numbers bundled in *n*-tuples like (*a*_{1}, *a*_{2}, *a*_{3}, ..., *a*_{n}). Unless otherwise specified, the components of vectors are all real numbers. A set of *n*-tuples is generally called a **manifold** in which nothing is defined, such as lengths, angles, and so on. But in this textbook, we do not deal with a general mathematical space, an *n*-dimensional manifold. Instead, we deal with only the standard space called the Euclid Space in which the distance and the angle are defined by familiar geometrical images or visualizations of objects in 3-Dimensional Mathematical Space, as depicted by the coordinate axes and a sphere in __Fig. 1-1__. Although we cannot visualize *n*-D spaces (*n* > 3), we infer higher dimensional mathematical spaces by logic.

__Fig. 1-1__

__Notational Conventions in this textbook__:

In some textbooks, vectors are denoted by an upper arrow: , or an upper bar: . In this textbooks, such notations may be adopted in situations, for directional line segments in geometry, but notations mainly used for vectors are:

an *italic* **boldface**: **a**, **b**, **c**, **x**, **y**, **z**, etc.

or

a ket-vector: , etc.

The components of row-vectors are written laterally and denoted by under bar or by bra-vectors as in

__a__ or = (*a*_{1}, *a*_{2}, *a*_{3}, ..., *a*_{n}).

In the above, the state of row-vectors are already indicated by the square bra direction in , so it may be redundant; no need for the under bar **_** inside, although you are free to write as . On the other, the components of col-vecters (column-vectors) are written in column form. Thus, col-vectors are denoted by *italic* **boldface** or by ket-vectors like

where ^{T} on the shoulder denotes the **transpose** operation to save line spaces, flipping rows and columns, indicating that it is actually written vertically. Note that each index in superscript on the components does not denote the power. It is used to distinguish col-vectors from row-vectors, or vice versa. If the power needs to be expressed, the parenthesis is used as in (*x*^{k})^{2}. When these distinctions are unnecessary, subscripts will be used.

Note that row-vectors are called 1-forms, and col-vectors are called 1-vectors in parlance of tensors in some textbooks. Each is defined by the relation < 1-form | 1-vector > = a number. That is,

row-vectors = bra-vectors = 1-forms ...in general, *p*-forms

col-vectors = ket-vectors = 1-vectors ...in general, *q*-vectors

—— Why do we need these designations? Because we need to distinguish an operator from an object (operand). But it will later turn out that operators and objects are interchangeable by transpose operation or flipping columns and rows. For example, let a ket-vector be . Then, , or vice versa: . They are said to be '**dual**' to each other. Operators act on objects and bring about some changes. A matrix acts as a representation of a 'linear' operator. ('Linearity' is defined and explained in §1-4.) Keep in mind also that we do not deal with continuous components or functions but only discrete components in these objects, row-vectors and col-vectors in Part I: Discrete Case. In Part II: Continuous Case, we deal with a function or Fourier transform as in , , which are extensively used in Quantum Mechanics. [The bra(c)ket formalism was invented and used by P.A.M. Dirac in his famous book of QM.]

__Reminder__:

1°. We consider algebraic *n*-tuple objects in the standard Euclid space in this textbook.

2°. We deal with row- and col-vectors having only discrete components in this textbook.