An Introduction to Linear Algebra
"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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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