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 50
... DIMENSIONALITY d(9Jl) of a linear manifold 9]! is the maximum value of r for which 5111 contains r linearly independent elements.T If no such maximum value exists, then d(EUl) = 00. The statement d(iUl) = r (where r > 0) means,. 1' If T1 ...
... DIMENSIONALITY d(9Jl) of a linear manifold 9]! is the maximum value of r for which 5111 contains r linearly independent elements.T If no such maximum value exists, then d(EUl) = 00. The statement d(iUl) = r (where r > 0) means,. 1' If T1 ...
Page 51
... dimensionality is said to be finite-dimensional. In contrast, if d(iUt) = 00 (i.e. if SIR contains arbitrarily large ... dimensionality of a linear manifold and the nature of its bases by showing that if a linear manifold has a basis ...
... dimensionality is said to be finite-dimensional. In contrast, if d(iUt) = 00 (i.e. if SIR contains arbitrarily large ... dimensionality of a linear manifold and the nature of its bases by showing that if a linear manifold has a basis ...
Page 52
... dimensionality now follows as a consequence of definitions and need not be taken as a separate postulate. Theorem 2.3.3. (Basis theorem for vector spaces) Let Q) be a non-null vector space of order n. Then 23 has finite dimensionality r ...
... dimensionality now follows as a consequence of definitions and need not be taken as a separate postulate. Theorem 2.3.3. (Basis theorem for vector spaces) Let Q) be a non-null vector space of order n. Then 23 has finite dimensionality r ...
Page 53
... dimensionality of the total vector space 23,, is n, and a basis of 5B,, is simply a set of n linearly independent vectors of order n. Since 53,, possesses a basis (E consisting of the n vectors e1,..., en, it follows that d(5B,,) = n ...
... dimensionality of the total vector space 23,, is n, and a basis of 5B,, is simply a set of n linearly independent vectors of order n. Since 53,, possesses a basis (E consisting of the n vectors e1,..., en, it follows that d(5B,,) = n ...
Page 59
... dimensionality. We are now in a position to show in what sense the study of linear manifolds can be replaced by that of vector spaces. Theorem 2.4.3. (Isomorphism theorem for linear manifolds) Every linear manifold SIR over 3, which has ...
... dimensionality. We are now in a position to show in what sense the study of linear manifolds can be replaced by that of vector spaces. Theorem 2.4.3. (Isomorphism theorem for linear manifolds) Every linear manifold SIR over 3, which has ...
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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