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. |
À l'intérieur du livre
Résultats 1-5 sur 54
Page 18
... assume that it is true for n - 1 , where n > 3 , and deduce that it is true for n . We may clearly assume that all the a's are distinct , for otherwise ( 1.4.5 ) is true trivially . Consider the determinant xn - 1 xn - 2 an - 1 an - 2 X ...
... assume that it is true for n - 1 , where n > 3 , and deduce that it is true for n . We may clearly assume that all the a's are distinct , for otherwise ( 1.4.5 ) is true trivially . Consider the determinant xn - 1 xn - 2 an - 1 an - 2 X ...
Page 28
... assume , without loss of generality , that a110 . In that case we subtract , for i = 2 , ... , n , a¿1 / α11 times the first row from the ith row in D and obtain · • a11 a12 0 b22 · ain ban D = 0 bn2 bij aij a11 where Hence b22 · bn2 ...
... assume , without loss of generality , that a110 . In that case we subtract , for i = 2 , ... , n , a¿1 / α11 times the first row from the ith row in D and obtain · • a11 a12 0 b22 · ain ban D = 0 bn2 bij aij a11 where Hence b22 · bn2 ...
Page 29
L. Mirsky. therefore , assume that a110 . If t1 , ... , to satisfy ( 1.6.1 ) , then ( 1.6.4 ) holds and therefore so does ( 1.6.3 ) . But b22 · ban 0 . bnn • bn2 · Hence , by ( 1.6.3 ) and the induction hypothesis , t2 = ... = t2 = 0 ...
L. Mirsky. therefore , assume that a110 . If t1 , ... , to satisfy ( 1.6.1 ) , then ( 1.6.4 ) holds and therefore so does ( 1.6.3 ) . But b22 · ban 0 . bnn • bn2 · Hence , by ( 1.6.3 ) and the induction hypothesis , t2 = ... = t2 = 0 ...
Page 30
... Assume that it holds for n - 1 , where n > 3. If now α11 = ... an - 1,1 tn = 0 , then the equations ( 1.6.7 ) are satisfied by t1 = 1 , t2 = ... = t2 = 0. If , however , a11 , ... , an - 1,1 do not all vanish , we may assume that a11 In ...
... Assume that it holds for n - 1 , where n > 3. If now α11 = ... an - 1,1 tn = 0 , then the equations ( 1.6.7 ) are satisfied by t1 = 1 , t2 = ... = t2 = 0. If , however , a11 , ... , an - 1,1 do not all vanish , we may assume that a11 In ...
Page 32
... Assume that it holds for n - 1 , where n > 3. We know that a11 0 ; and subtracting , for r = 2 , ... , n , a , 1 / a11 times the first row from the rth row , we obtain D = a11 D ' , where D ' b22 = 3 bn2 ar1 a18 brs ars and ban bnn ( r ...
... Assume that it holds for n - 1 , where n > 3. We know that a11 0 ; and subtracting , for r = 2 , ... , n , a , 1 / a11 times the first row from the rth row , we obtain D = a11 D ' , where D ' b22 = 3 bn2 ar1 a18 brs ars and ban bnn ( r ...
Autres éditions - Tout afficher
Expressions et termes fréquents
A₁ algebra assertion automorphism b₁ basis bilinear form bilinear operator canonical forms characteristic roots characteristic vectors coefficients columns commute complex numbers convergent coordinates Deduce defined denote determinant diagonal form diagonal matrix dimensionality E-operations edition equal equivalence EXERCISE exists follows functions geometry given Hence hermitian form hermitian matrix identity implies inequality integers invariant space isomorphic linear equations linear manifold linear transformation linearly independent matrix group minimum polynomial multiplication non-singular linear transformation non-singular matrix non-zero numbers nxn matrix obtain orthogonal matrix positive definite positive semi-definite possesses problems proof of Theorem prove quadratic form quadric rank relation represented respect result rotation S-¹AS satisfies scalar Show similar singular skew-symmetric matrix solution square matrix suppose symmetric matrix t₁ theory tions triangular unique unit element unitary matrix values vector space view of Theorem write x₁ xTAx y₁ zero