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 18
... independent of x , as may be seen by comparing the degrees of the two sides of the equation . Now , by ( 1.4.1 ) , the coefficient of x ^ -1 in ƒ ( x ) is equal to απ n - 2 2-2 an · a2 an · 1 1 which , by the induction hypothesis , is ...
... independent of x , as may be seen by comparing the degrees of the two sides of the equation . Now , by ( 1.4.1 ) , the coefficient of x ^ -1 in ƒ ( x ) is equal to απ n - 2 2-2 an · a2 an · 1 1 which , by the induction hypothesis , is ...
Page 33
... independent proof has been given by G. B. Price , Proc . Amer . Math . Soc . 2 ( 1951 ) , 497–502 . prove that An - An - 1 = ani 91 I , § 1.6 TWO SPECIAL THEOREMS ON LINEAR EQUATIONS 33.
... independent proof has been given by G. B. Price , Proc . Amer . Math . Soc . 2 ( 1951 ) , 497–502 . prove that An - An - 1 = ani 91 I , § 1.6 TWO SPECIAL THEOREMS ON LINEAR EQUATIONS 33.
Page 47
... 1 ) X = −X = −X . This , together with conditions ( vi ) and ( vii ) , implies — ( − X ) = ( -1 ) { ( − 1 ) X } = { ( −1 ) ( − 1 ) } X = 1X = X. elements X1 , ... , X1⁄2 are LINEARLY INDEPENDENT . II , § 2.2 47 LINEAR MANIFOLDS.
... 1 ) X = −X = −X . This , together with conditions ( vi ) and ( vii ) , implies — ( − X ) = ( -1 ) { ( − 1 ) X } = { ( −1 ) ( − 1 ) } X = 1X = X. elements X1 , ... , X1⁄2 are LINEARLY INDEPENDENT . II , § 2.2 47 LINEAR MANIFOLDS.
Page 48
... LINEARLY INDEPENDENT . k It should be observed that linear dependence or independence is a property of the unordered set of elements X1 , ... , X. An analogous remark applies to a number of statements below . For vectors over ...
... LINEARLY INDEPENDENT . k It should be observed that linear dependence or independence is a property of the unordered set of elements X1 , ... , X. An analogous remark applies to a number of statements below . For vectors over ...
Page 49
... linearly independent , then so are X1 , ... , X. ( ii ) Show that if X1 , ... , X are linearly dependent and Y1 , ... , Y are any elements , then X1 , ... , X , Y1 , ... , Y , are linearly dependent . DEFINITION 2.3.2 . A BASIS of a linear ...
... linearly independent , then so are X1 , ... , X. ( ii ) Show that if X1 , ... , X are linearly dependent and Y1 , ... , Y are any elements , then X1 , ... , X , Y1 , ... , Y , are linearly dependent . DEFINITION 2.3.2 . A BASIS of a linear ...
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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