An Introduction to Linear AlgebraClarendon Press, 1955 - 433 pages |
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Page 52
... linearly dependent if s > r . On the other hand , there exists a set of r linearly independent elements , namely E ... independent elements in M. If Xe M , then , by the definition of r , the r + 1 elements X , E1 , ... , E , are ...
... linearly dependent if s > r . On the other hand , there exists a set of r linearly independent elements , namely E ... independent elements in M. If Xe M , then , by the definition of r , the r + 1 elements X , E1 , ... , E , are ...
Page 54
... linearly independent elements in a finite - dimensional linear manifold M is part of a basis of M. k k Let d ( m ) = r and let X1 , ... , X be a set of linearly independent elements in M , so that k < r . If kr , then , by Theorem 2.3.2 ...
... linearly independent elements in a finite - dimensional linear manifold M is part of a basis of M. k k Let d ( m ) = r and let X1 , ... , X be a set of linearly independent elements in M , so that k < r . If kr , then , by Theorem 2.3.2 ...
Page 320
... linearly independent we mean , of course , that the relation ... α1 A1 + ... + αk Ak = 0 implies α1 = = αk = 0. A set of nxn matrices contains at most n2 linearly independent ones . If k is the maximum number of linearly independent ...
... linearly independent we mean , of course , that the relation ... α1 A1 + ... + αk Ak = 0 implies α1 = = αk = 0. A set of nxn matrices contains at most n2 linearly independent ones . If k is the maximum number of linearly independent ...
Table des matières
PART | 3 |
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 dimensionality E-operations equal equivalence EXERCISE exists follows functions given Hence hermitian form hermitian matrix identity implies inequality integers 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 non-zero nxn matrix obtain orthogonal matrix positive definite possesses proof of Theorem prove quadratic form quadric rank reduces 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 w₁ write x₁ y₁ zero α₁