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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 x1, x2, ..., 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 a11, a12, ..., 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 b1, b2, ..., 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 c1, c2, ..., cn which when substituted for the unknowns x1, x2, ..., 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
2x1 + 3x2 = 5, 2x1 + 3x2 = 6 (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 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
2x1 + 3x2 = 0, 4x1 + 6x2 = 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 n2 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 a1 the number of the row in which the element appears, the indices of the element will be a1, 1. Similarly, for the second factor we can choose the element appearing in the second column; then its indices will be a2, 2, where a2 is the number of the row in which the element appears, and so on. Thus, the indices a1, a2, ..., an 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 a1, a2, ..., an are all different and represent some permutation of the numbers 1, 2, ..., n.
By an inversion in the sequence a1, a2, ..., 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(a1, a2, ..., 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 a1, a2, ..., 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:
a11x1 + a12x2 = b1, a21x1 + a22x2 = b2.
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 a21a12a43a34, 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 a41a32a13a24, 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 ai >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. a23a31a42a56a14a65,
2. a32a43a14a51a66a25 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. a11a32a23a44, a41a12a23a34, a31a42a23a14.
Excerpted from An Introduction to the Theory of Linear Spaces by Georgi E. Shilov, Richard A. Silverman. Copyright © 1989 Richard A. Silverman. Excerpted by permission of Dover Publications, Inc..
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Overview
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.