An Introduction to Linear AlgebraCourier Corporation, 3 déc. 2012 - 464 pages "The straight-forward clarity of the writing is admirable." — American Mathematical Monthly. This work provides an elementary and easily readable account of linear algebra, in which the exposition is sufficiently simple to make it equally useful to readers whose principal interests lie in the fields of physics or technology. The account is self-contained, and the reader is not assumed to have any previous knowledge of linear algebra. Although its accessibility makes it suitable for non-mathematicians, Professor Mirsky's book is nevertheless a systematic and rigorous development of the subject. Part I deals with determinants, vector spaces, matrices, linear equations, and the representation of linear operators by matrices. Part II begins with the introduction of the characteristic equation and goes on to discuss unitary matrices, linear groups, functions of matrices, and diagonal and triangular canonical forms. Part II is concerned with quadratic forms and related concepts. Applications to geometry are stressed throughout; and such topics as rotation, reduction of quadrics to principal axes, and classification of quadrics are treated in some detail. An account of most of the elementary inequalities arising in the theory of matrices is also included. Among the most valuable features of the book are the numerous examples and problems at the end of each chapter, carefully selected to clarify points made in the text. |
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Page 2
... is almost obvious. To prove (i), suppose that for two different choices of (1/1,...,v,,)—say (oz1,..., a,,) and (B1,...,}3,,)— (v,,1,..., 11,“) is the same arrangement, i.e. ((Xkl,.--, Gk") = (Bkp.--,Bkn), and SO (Xkl = Bk', u.' U)'.
... is almost obvious. To prove (i), suppose that for two different choices of (1/1,...,v,,)—say (oz1,..., a,,) and (B1,...,}3,,)— (v,,1,..., 11,“) is the same arrangement, i.e. ((Xkl,.--, Gk") = (Bkp.--,Bkn), and SO (Xkl = Bk', u.' U)'.
Page 11
... suppose that D' denotes the determinant obtained when is times the sth row is added to the rth row in D. Assuming that r < s we have an ' - - am a,1-Hca,1 . . . a,.,,—{-Icam D' = . . . . . I I O a a n1 nn Hence, by Theorem 1.2.5 (as ...
... suppose that D' denotes the determinant obtained when is times the sth row is added to the rth row in D. Assuming that r < s we have an ' - - am a,1-Hca,1 . . . a,.,,—{-Icam D' = . . . . . I I O a a n1 nn Hence, by Theorem 1.2.5 (as ...
Page 26
... suppose the minor M to consist of those elements of D which belong to the rows with suffixes r1,...,rk and to the columns with suflixes s1,...,sk (where r1 < < rk and 81 < < sk). We write r1+...+rk+s1+...+s,, = t. Our aim is to reduce ...
... suppose the minor M to consist of those elements of D which belong to the rows with suffixes r1,...,rk and to the columns with suflixes s1,...,sk (where r1 < < rk and 81 < < sk). We write r1+...+rk+s1+...+s,, = t. Our aim is to reduce ...
Page 27
... Suppose it holds for n-— 1, where n 2 2; we shall then show that it also holds for n. Let (1.6.2) be satisfied. If a11 = = 1' It must, of course, be remembered that a;,- does not necessarily stand in the ith row and jth column of 9 ...
... Suppose it holds for n-— 1, where n 2 2; we shall then show that it also holds for n. Let (1.6.2) be satisfied. If a11 = = 1' It must, of course, be remembered that a;,- does not necessarily stand in the ith row and jth column of 9 ...
Page 30
... later chapters. THEOREM 1.6.3. If the polynomial f(x) = cox"+c1x""1+...+c,,_1x+en vanishes for n+1 distinct values of x, then it vanishes identically. Let a:1,...,:z:n+1 be distinct numbers, and suppose that f(x1) : 30 DETERMINANTS I,§ 1.6.
... later chapters. THEOREM 1.6.3. If the polynomial f(x) = cox"+c1x""1+...+c,,_1x+en vanishes for n+1 distinct values of x, then it vanishes identically. Let a:1,...,:z:n+1 be distinct numbers, and suppose that f(x1) : 30 DETERMINANTS I,§ 1.6.
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algebra assertion assume automorphism basis bilinear form bilinear operator canonical forms characteristic polynomial characteristic roots characteristic vectors coefiicients commute complement complex numbers convergent coordinates Deduce defined DEFINITION denote determinant diagonal elements diagonal form diagonal matrix dimensionality E-operations edition equal EXERCISE field find finite fixed follows functions geometry Hence hermitian form hermitian matrix identity implies inequality infinite integers inverse isomorphic linear combination linear equations linear manifold linear transformation linearly independent matrix group matrix of order minimum polynomial multiplication non-singular linear transformation non-singular matrix numbers obtain orthogonal matrix permutation positive semi-definite possesses problems proof of Theorem prove quadratic form quadric rank real symmetric reduces representation represented respect result rotation satisfies scalar Show similar singular skew-symmetric matrix solution specified square matrix suppose symmetric matrix Theorem theory tions unique unit element unitary matrix values vanish variables vector space view of Theorem write zero