An Introduction to the Theory of Linear Spaces

Georgi E. Shilov, Richard A. Silverman

• Verlag: Dover Publications
• Jahr: 2012
• ISBN:
  9780486139432
• Thema: Mathematics
• Thema: Linear Algebra

This introduction to linear algebra and functional analysis offers a clear expository treatment, viewing algebra, geometry, and analysis as parts of an integrated whole rather than separate subjects. All abstract ideas receive a high degree of motivation, and numerous examples illustrate many different fields of mathematics. Abundant problems include hints or answers.

1

DETERMINANTS

I. Systems of Linear Equations

In this and the next two chapters, we shall study systems of linear equations. In the most general case, such a system has the form

(1)

Here x₁, x₂, . . . , x_n denote the unknowns which are to be determined. (Note that we do not necessarily assume that the number of unknowns equals the number of equations.) The quantities a₁₁, a₁₂, . . . , a_kn are called the coefficients of the system. The first index of a coefficient indicates the number of the equation in which the coefficient appears, while the second index indicates the number of the unknown with which the coefficient is associated.¹ The quantities b₁, b₂, . . . , b_k appearing in the right-hand side of (1) are called the constant terms of the system; like the coefficients, they are assumed to be known. By a solution of the system (1) we mean any set of numbers c₁, c₂, . . . , c_n which when substituted for the unknowns x₁, x₂, . . . , x_n, turns all the equations of the system into identities.²

Not every system of linear equations of the form (1) has a solution. For example, the system

(2)

obviously has no solution at all. Indeed, whatever numbers c₁, c₂ we substitute in place of the unknowns x₁, x₂, the left-hand sides of the equations of the system (2) are the same, while the right-hand sides are different. Therefore, no such substitution can simultaneously convert both equations of the system into identities.

A system of equations of the form (1) which has at least one solution is called compatible; a system which does not have solutions is called incompatible. A compatible system can have one solution or several solutions. In the latter case, we distinguish the solutions by indicating the number of the solution by a superscript in parentheses; for example, the first solution will be denoted by

,

, . . . ,

, the second solution by

,

, . . . ,

, and so on. The solutions

,

, . . . ,

and

,

, . . . ,

are considered to be distinct if at least one of the numbers

does not coincide with the corresponding numbers

(i = 1, 2, ..., n). For example, the system

(3)

has the distinct solutions

(and also infinitely many other solutions). If a compatible system has a unique solution, the system is called determinate; if a compatible system has at least two different solutions, it is called indeterminate.

We can now formulate the basic problems which arise in studying the system (1):

1. To ascertain whether the system (1) is compatible or incompatible;
2. If the system (1) is compatible, to ascertain whether it is determinate;
3. If the system (1) is compatible and determinate, to find its unique solution;
4. If the system (1) is compatible and indeterminate, to describe the set of all its solutions.

Th...