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 21
... COMPLEMENT ) D ( r1 , ... , rk | 81 , ... , 8 % ) of the minor D ( r1 , ... , * | 81 , ... , 8x ) in a determinant D is defined as Ď ( r1 , ... , rk | 81 , ... , Sk ) = ( −1 ) 31 + ... + r + 81 + ...... . + 8 % D ( rk + 1 , ·· ...
... COMPLEMENT ) D ( r1 , ... , rk | 81 , ... , 8 % ) of the minor D ( r1 , ... , * | 81 , ... , 8x ) in a determinant D is defined as Ď ( r1 , ... , rk | 81 , ... , Sk ) = ( −1 ) 31 + ... + r + 81 + ...... . + 8 % D ( rk + 1 , ·· ...
Page 57
... complements without having yet considered the question of their existence . However , the theorem just proved enables us to settle this question . THEOREM 2.3.9 . Every subspace of Vn possesses a complement . n Let U be a subspace of B ...
... complements without having yet considered the question of their existence . However , the theorem just proved enables us to settle this question . THEOREM 2.3.9 . Every subspace of Vn possesses a complement . n Let U be a subspace of B ...
Page 68
... COMPLEMENT of U is the set of all vectors in V , which are orthogonal to every vector in U. ‡ n n EXERCISE 2.5.7 . Show that the orthogonal complement of V1 is the null space , and that the orthogonal complement of the null space is Vn ...
... COMPLEMENT of U is the set of all vectors in V , which are orthogonal to every vector in U. ‡ n n EXERCISE 2.5.7 . Show that the orthogonal complement of V1 is the null space , and that the orthogonal complement of the null space is Vn ...
Page 69
... complement of U ' is spanned by X1 , ... , X , and so is identical with U. Moreover , d ( u ' ) = n ― r , and therefore d ( ) + d ( U ' ) = n . THEOREM 2.5.8 . The orthogonal complement of U is a complement of U. = = Let U ' denote the ...
... complement of U ' is spanned by X1 , ... , X , and so is identical with U. Moreover , d ( u ' ) = n ― r , and therefore d ( ) + d ( U ' ) = n . THEOREM 2.5.8 . The orthogonal complement of U is a complement of U. = = Let U ' denote the ...
Page 71
... complement of U. 4 21. The subspace U of V , is spanned by the vectors ( 1,0 , — 1 , 2 ) and ( -1,1,1,0 ) . Show that the orthogonal complement l ' of U is the set of vectors of the form ( a - 2ß , -2ß , a , ß ) , where a , ẞ are ...
... complement of U. 4 21. The subspace U of V , is spanned by the vectors ( 1,0 , — 1 , 2 ) and ( -1,1,1,0 ) . Show that the orthogonal complement l ' of U is the set of vectors of the form ( a - 2ß , -2ß , a , ß ) , where a , ẞ are ...
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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