Linear algebra problems book docx

00 64431 co co > [...]... of algebraical entities The matrix written above, is a matrix with (i) m= n, is associated with the determinant *2n nl (ii) with the form 2 2 r-l *,** of degree 2 in the n variables x v x 2 , , x n with the bilinear form (iii) I I ars^ys x l9 , x n and y l9 , y n with the Hermitian form in the 2n variables (iv) \ \ 2 2 Or^r^ r=l s=l where (v) rr a and fs are conjugate complex numbers; with the linear. .. does not contain the number Most of the propositions the work field is 1 rxs, set Notice that the set the numbers rs, in this contains | book presuppose that what particular carried out within a field of numbers; usually of little consequence In the early part of the book this aspect of the matter need not be emphasized: in some of the later chapters the essence of the theorem is that all the operations... of theory that is of constant application in other branches of mathematics is relatively small, and it is this restricted range that the book covers 1.2 Determinants of the second and third orders Determinants are, in origin, closely connected with the solu- tion of linear equations Suppose that the two equations a^x+b^j are satisfied = a 2 x+b 2 y 0, by a pair of numbers x and from zero Then y, = one... either J, or irrational, such as 77 or V3 Finally, with the solution of the quadratic equation, x is permitted to be a complex number, such as 2-\-3i rational, such as 3 J or The numbers used in this book may be either real or complex and we shall assume that readers have studied, to a greater or a lesser extent, the precise definitions of these numbers and the rules governing their addition, subtraction,... matrix and of its associated forms are closely knit together The plan of expounding these theories that I have adopted is, roughly, this: Part I develops properties of the determinant; Part II develops the algebra of matrices, referring may back to Part I for any result about determinants that be needed; Part III develops the theory of the other associated forms CHAPTER I ELEMENTARY PROPERTIES OF DETERMINANTS...PART I PRELIMINARY NOTE; CHAPTERS ON DETERMINANTS PRELIMINARY NOTE Number 1 In its initial stages algebra is little more than a generalization of elementary arithmetic It deals only with the positive can all remember the type of problem integers, 1, 2, 3, that began 'let We x be the number of eggs',... three 63 referred to as a 'determinant of order three' or a 'determinant of the third order 5 The term a x 62 c3 is referred to as 'the leading diagonal' 1.3 It is thus suggested that, associated with n linear equations in n variables, say = z = iX+b l y+ +k l z 0, a 2 x+b 2 y+ +k 2 0, ELEMENTARY PROPERTIES OF DETERMINANTS there is a certain function of the coefficients which must be the equations... all the definitions lead to the same result in the end The only particular merit we claim for our own definition is that it is easily reconcilable with any of the others, and so makes reference to other books a simple matter Properties of determinants of order three As we have seen in 1.2, the determinant 1.5 b ( stands for the expression (1) The following facts are all but self-evident: (I) The expression... order 6 sum definition, for n, n of the 2 determinants corresponding to the ways of choosing one letter from each column; in particular, is the sum of the four determinants bi a2 62 This again is is the algebraic obvious from the definition; for sum of the 2" summations typified by taking one term from each bracket 16 ELEMENTARY PROPERTIES OF DETERMINANTS THEOREM The value of a determinant is unaltered... 6 e 2 -f- c 2 / 2 2bcef e e a d 2 d b c c is / / 2cafd V/12 Prove that the expansion of 1 1 1 y l 2 2/ is /1 3 (Harder.) Express the determinant bed x-\-c d a b x-\~c as a product of factors HINT Two linear factors x + a+c(b-{-d) and one quadratic 14 From a determinant A, of order formed by taking c{ = Prove that A' Cj-f- Ac 2 , Ca = c 2 -f^c3 , 3 4, = factor a new determinant A' c3 + vc l9 ci = is . 5,000. OSMANIA UNIVERSITY LIBRARY Call No. 5 1 ^ 1 F 3 7 A Accession No. Author This book should be returned on or before the date last marked below.