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 iii
... quadrics to principal axes , rotations in the plane and in space , and the classification of quadrics under the projective and affine groups . The theory of matrices gives rise to many striking inequalities.
... quadrics to principal axes , rotations in the plane and in space , and the classification of quadrics under the projective and affine groups . The theory of matrices gives rise to many striking inequalities.
Page viii
... QUADRATIC FORMS XII . BILINEAR , QUADRATIC , AND HERMITIAN FORMS 12.1 . Operators and forms of the bilinear and quadratic types 12.2 . Orthogonal reduction to diagonal form 353 362 12.3 . General reduction to diagonal form 367 12.4 ...
... QUADRATIC FORMS XII . BILINEAR , QUADRATIC , AND HERMITIAN FORMS 12.1 . Operators and forms of the bilinear and quadratic types 12.2 . Orthogonal reduction to diagonal form 353 362 12.3 . General reduction to diagonal form 367 12.4 ...
Page 19
... quadratic polynomial x2 + μx + v is μ2 - 4v . We now resume our discussion of the general theory of determi- nants . THEOREM 1.4.2 . With the same notation as in Theorem 1.4.1 we have for rs , n Σark Ask 0 , k = 1 Σακάκο k = 1 A ks ** 0 ...
... quadratic polynomial x2 + μx + v is μ2 - 4v . We now resume our discussion of the general theory of determi- nants . THEOREM 1.4.2 . With the same notation as in Theorem 1.4.1 we have for rs , n Σark Ask 0 , k = 1 Σακάκο k = 1 A ks ** 0 ...
Page 34
... quadratic and four linear factors . 3. By squaring the determinant b с 0 α 0 C 0 a b bε + c3 ab са show that ab c2 + a2 bc = 4a2bc2 . са bc a2 + b2 4. Use Laplace's expansion to show that α -b -a b b α -b -a -d с -d с d с d с = 4 ( a2 + ...
... quadratic and four linear factors . 3. By squaring the determinant b с 0 α 0 C 0 a b bε + c3 ab са show that ab c2 + a2 bc = 4a2bc2 . са bc a2 + b2 4. Use Laplace's expansion to show that α -b -a b b α -b -a -d с -d с d с d с = 4 ( a2 + ...
Page 62
... quadratic operators in Chapters IV and XII . 2.5 . Inner products and orthonormal bases 2.5.1 . Let a system of rectangular coordinates be introduced in three - dimensional space , and let ( x1 , x2 , 3 ) , ( Y1 , Y2 , Y3 ) be the ...
... quadratic operators in Chapters IV and XII . 2.5 . Inner products and orthonormal bases 2.5.1 . Let a system of rectangular coordinates be introduced in three - dimensional space , and let ( x1 , x2 , 3 ) , ( Y1 , Y2 , Y3 ) be the ...
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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