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 42
... LINEAR COMBINATION of the vectors x1,..., xk. If a vector X is equal to some linear combirmtion of X1,..., xk, it is said to be expressible linearly in terms of, or to depend linearly on, X1,..., xk. It is obvious that a linear ...
... LINEAR COMBINATION of the vectors x1,..., xk. If a vector X is equal to some linear combirmtion of X1,..., xk, it is said to be expressible linearly in terms of, or to depend linearly on, X1,..., xk. It is obvious that a linear ...
Page 48
... linear manifold fill is spanned by the elements X1,..., X,., Y1,..., Y8, and ifeach Y is a linear combination of the X's, show that W1 is spanned by X,,...,X,. 2.3. Linear dependence and bases 2.3.1. Let T? again denote a linear ...
... linear manifold fill is spanned by the elements X1,..., X,., Y1,..., Y8, and ifeach Y is a linear combination of the X's, show that W1 is spanned by X,,...,X,. 2.3. Linear dependence and bases 2.3.1. Let T? again denote a linear ...
Page 49
... linearly dependent. DEFINITION 2.3.2. A BASIS of a linear manifold 93? is a finite ordered set of elements X1,..., X k e EUI having the property that every X e SUI is expressible as a unique linear combination of X1,..., X ,,.T Thus the ...
... linearly dependent. DEFINITION 2.3.2. A BASIS of a linear manifold 93? is a finite ordered set of elements X1,..., X k e EUI having the property that every X e SUI is expressible as a unique linear combination of X1,..., X ,,.T Thus the ...
Page 58
... linear manifolds are, of course, said to be isomorphic if there exists an isomorphism between them. EXERCISE 2.4.1. Show ... linear combination of some elements in SIR and the same linear combination of the corresponding elements in SIR ...
... linear manifolds are, of course, said to be isomorphic if there exists an isomorphism between them. EXERCISE 2.4.1. Show ... linear combination of some elements in SIR and the same linear combination of the corresponding elements in SIR ...
Page 67
... linearly independent vectors. The procedure by means of which this object is achieved—known as Schmz'dt's ... linear combination of X1,..., xk in which xk has the coefficient 1. Since X1,...,Xk are linearly independent it ...
... linearly independent vectors. The procedure by means of which this object is achieved—known as Schmz'dt's ... linear combination of X1,..., xk in which xk has the coefficient 1. Since X1,...,Xk are linearly independent it ...
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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