An Introduction to Linear AlgebraClarendon Press, 1972 - 440 pages |
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Page 49
... basis if every X = M is expressible in the form X = α1X1 + ... + αxXx › where a , ... , a are uniquely determined . ‡ If a basis exists in M , then it furnishes M with a ' coordinate system ' , since we may regard the scalars a1 ...
... basis if every X = M is expressible in the form X = α1X1 + ... + αxXx › where a , ... , a are uniquely determined . ‡ If a basis exists in M , then it furnishes M with a ' coordinate system ' , since we may regard the scalars a1 ...
Page 51
... basis consisting of r elements , then the dimensionality of that manifold is r , and conversely . Theorem 2.3.2 . ( Basis theorem for linear manifolds ) Let M be a linear manifold . ( i ) If M has a basis of r elements , then d ( m ) ...
... basis consisting of r elements , then the dimensionality of that manifold is r , and conversely . Theorem 2.3.2 . ( Basis theorem for linear manifolds ) Let M be a linear manifold . ( i ) If M has a basis of r elements , then d ( m ) ...
Page 53
... basis of V , if and only if ad bc . COROLLARY 1. The dimensionality of the total vector space Vn is n , and a basis of Vn is simply a set of n linearly independent vectors of order n . n Since possesses a basis & consisting of the n ...
... basis of V , if and only if ad bc . COROLLARY 1. The dimensionality of the total vector space Vn is n , and a basis of Vn is simply a set of n linearly independent vectors of order n . n Since possesses a basis & consisting of the n ...
Table des matières
DETERMINANTS VECTORS MATRICES | 1 |
VECTOR SPACES AND LINEAR MANIFOLDS | 39 |
THE ALGEBRA OF MATRICES | 72 |
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Expressions et termes fréquents
A₁ algebra assertion automorphism B₁ basis bilinear form bilinear operator C₁ canonical forms characteristic roots characteristic vectors coefficients columns commute complex numbers convergent coordinates Deduce defined denote determinant diagonal form diagonal matrix E-operations equal equivalence EXERCISE exists follows functions given Hence hermitian form hermitian matrix identity implies inequality integers invariant space isomorphic linear equations linear manifold linear transformation linearly independent matrix group matrix of order minimum polynomial multiplication non-singular linear transformation non-singular matrix non-zero numbers nxn matrix obtain orthogonal matrix positive definite positive semi-definite possesses 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 tion triangular unique unit element unitary matrix values vector space view of Theorem write x₁ xTAx y₁ zero α₁