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#### An Introduction to the Theory of Linear Spaces

**By Georgi E. Shilov, Richard A. Silverman**

**Dover Publications, Inc.**

**Copyright © 1989 Richard A. Silverman**

All rights reserved.

ISBN: 978-0-486-13943-2

All rights reserved.

ISBN: 978-0-486-13943-2

CHAPTER 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(1)*

Here *x*1, *x*2, ..., *xn* 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 *a*11, *a*12, ..., *akn* 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. The quantities *b*1, *b*2, ..., *bk* 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 *c*1, *c*2, ..., *cn* which when substituted for the unknowns *x*1, *x*2, ..., *xn*, turns all the equations of the system into identities.

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

2*x*1 + 3*x*2 = 5, 2*x*1 + 3*x*2 = 6 *(2)*

obviously has no solution at all. Indeed, whatever numbers *c*1, *c*2 we substitute in place of the unknowns *x*1, *x*2, 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 *c*(1)1, *c*(1)2, ..., *c*(1)*n*, the second solution by *c*(2)1, *c*(2)2, ..., *c*(2)*n*, and so on. The solutions *c*(1)1, *c*(1)2, ..., *c*(1)*n* and *c*(2)1, *c*(2)2, ..., *c*(2)*n* are considered to be *distinct* if at least one of the numbers *c*(1)1 does not coincide with the corresponding numbers *c*(2)1 (*i* = 1, 2, ..., *n*). For example, the system

2*x*1 + 3*x*2 = 0, 4*x*1 + 6*x*2 = 0 *(3)*

has the distinct solutions

*c*(1)1 = *c*(1)2 = 0 and *c*(2)1 = 3, *c*(2)2 = -2

(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.*

The basic mathematical tool for studying linear systems is *the theory of determinants*, which we consider next.

**2. Determinants of Order n**

**2.1** Suppose that we are given a *square matrix*, i.e., an array of *n*2 numbers *aij* (*i,j* = 1, 2, ..., *n*):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(4)*

The number of rows and columns of the matrix (4) is called its *order*. The numbers *aij* are called the *elements* of the matrix. The first index indicates the row and the second index the column in which *aij* appears.

Consider any product of *n* elements which appear in different rows and different columns of the matrix (4), i.e., a product containing *just one element from each row and each column*. Such a product can be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(5)*

Actually, for the first factor we can always choose the element appearing in the first column of the matrix (4); then, if we denote by *a*1 the number of the row in which the element appears, the indices of the element will be *a*1, 1. Similarly, for the second factor we can choose the element appearing in the second column; then its indices will be *a*2, 2, where *a*2 is the number of the row in which the element appears, and so on. Thus, the indices *a*1, *a*2, ..., *a*n are the numbers of the rows in which the factors of the product (5) appear, when we agree to write the column indices in increasing order. Since, by hypothesis, the elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ..., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] appear in *different* rows of the matrix (4), one from each row, then the numbers *a*1, *a*2, ..., *an* are all different and represent some permutation of the numbers 1, 2, ..., *n*.

By an *inversion* in the sequence *a*1, *a*2, ..., *an*, we mean an arrangement of two indices such that the larger index comes before the smaller index. The total number of inversions will be denoted by *N*(*a*1, *a*2, ..., *aN*). For example, in the permutation 2, 1, 4, 3, there are two inversions (2 before 1, 4 before 3), so that

*N*(2, 1, 4, 3) = 2.

In the permutation 4, 3, 1, 2, there are five inversions (4 before 3, 4 before 1, 4 before 2, 3 before 1, 3 before 2), so that

*N*(4, 3, 1, 2) = 5

If the number of inversions in the sequence *a*1, *a*2, ..., *an* is even, we put a plus sign before the product (5); if the number is odd, we put a minus sign before the product. In other words, we agree to write in front of each product of the form (5) the sign determined by the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The total number of products of the form (5) which can be formed from the elements of a given matrix of order *n* is equal to the total number of permutations of the numbers 1, 2, ..., *n*. As is well known, this number is equal to *n*!.

We now introduce the following definition:

*By the determinant D of the matrix* (4) *is meant the algebraic sum of the n! products of the form* (5), *each of which is preceded by the sign determined by the rule just given, i.e.,*

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(6)*

Henceforth, the products of the form (5) will be called the *terms* of the determinant. The elements *aij* of the matrix (4) will be called the *elements* of the determinant. We denote the determinant corresponding to the matrix (4) by one of the following symbols:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] *(7)*

For example, we obtain the following expressions for the determinants of orders two and three:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now indicate the role of determinants in solving systems of linear equations, by considering the example of a system of two equations in two unknowns:

*a*11*x*1 + *a*12*x*2 = *b*1, *a*21*x*1 + *a*22*x*2 = *b*2.

Eliminating one of the unknowns in the usual way, we can easily obtain the formulas

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

assuming that these ratios have nonvanishing denominators. The numerators and denominators of the ratios can be represented by the second-order determinants

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It turns out that similar formulas hold for the solutions of systems with an arbitrary number of unknowns (Sec. 7).

**2.2.** The rule for determining the sign of a given term of a determinant can be formulated somewhat differently, in geometric terms. Corresponding to the enumeration of elements in the matrix (4), we can distinguish two natural positive directions: from left to right along the rows, and from top to bottom along the columns. Moreover, the slanting lines joining any two elements of the matrix can be furnished with a direction: we shall say that the line segment joining the element *aij* with the element *akl* has *positive slope* if its right endpoint lies lower than its left endpoint, and that it has *negative slope* if its right endpoint lies higher than its left endpoint. Now imagine that in the matrix (4) we draw all the segments with *negative* slope joining pairs of elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ..., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the product (5). Then we put a plus sign before the product (5) if the number of all such segments is even, and a minus sign if the number is odd.

For example, in the case of a fourth-order matrix, a plus sign must be put before the product *a*21*a*12*a*43*a*34, since there are two segments of negative slope joining the elements of this product:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, a minus sign must be put before the product *a*41*a*32*a*13*a*24, since in the matrix there are five segments of negative slope joining these elements:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In these examples, the number of segments of negative slope joining the elements of a given term equals the number of *inversions* in the order of the first indices of the elements appearing in the term. In the first example, the sequence 2, 1, 4, 3 of first indices has two inversions; in the second example, the sequence 4, 3, 1, 2 of first indices has five inversions.

We now show that *the second definition of the sign of a term in a determinant is equivalent to the first*. To show this, it suffices to prove that the number of inversions in the sequence of first indices of a given term (with the second indices in natural order) is always equal to the number of segments of negative slope joining the elements of the given term in the matrix. But this is almost obvious, since the presence of a segment of negative slope joining the elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] means that *a*i >*aj* for *i*< *j*, i.e., there is an inversion in the order of the first indices.

** Problem 1.** With what sign do the terms

1. *a*23*a*31*a*42*a*56*a*14*a*65,

2. *a*32*a*43*a*14*a*51*a*66*a*25 appear in the determinant of order 6?

*Ans.* (a) +, (b) +.

** Problem 2.** Write down all the terms appearing in the determinant of order 4 which have a minus sign and contain the factor a23.

*Ans. a*11*a*32*a*23*a*44, *a*41*a*12*a*23*a*34, *a*31*a*42*a*23*a*14.

*(Continues...)*

Excerpted fromAn Introduction to the Theory of Linear SpacesbyGeorgi E. Shilov, Richard A. Silverman. Copyright © 1989 Richard A. Silverman. Excerpted by permission of Dover Publications, Inc..

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