An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 90
... follows by Theorem 1.5.1 that ( B * A * ) 1j = { ( AB ) * } ij and this is , in fact , our assertion . ( i , j = 1 , ... , n ) , An alternative argument of the now familiar kind runs as follows . Rewriting ( 3.5.3 ) with A - xI , B - I ...
... follows by Theorem 1.5.1 that ( B * A * ) 1j = { ( AB ) * } ij and this is , in fact , our assertion . ( i , j = 1 , ... , n ) , An alternative argument of the now familiar kind runs as follows . Rewriting ( 3.5.3 ) with A - xI , B - I ...
Page 150
... follows . Let L be a linear mapping † of V , into Vm Denote by U the vector space of vectors in Vm which are images of vectors in B , and by ' the vector space of vectors in V , which map into the zero vector of Vm . Then d ( u ' ) n ...
... follows . Let L be a linear mapping † of V , into Vm Denote by U the vector space of vectors in Vm which are images of vectors in B , and by ' the vector space of vectors in V , which map into the zero vector of Vm . Then d ( u ' ) n ...
Page 159
... follows that the same is true of the rows of AB . Hence R ( AB ) ≤ r = R ( A ) , and when r = n this inequality holds trivially . The proof is now completed in the same way as in ( i ) . ( iii ) A longer but more direct proof is as follows ...
... follows that the same is true of the rows of AB . Hence R ( AB ) ≤ r = R ( A ) , and when r = n this inequality holds trivially . The proof is now completed in the same way as in ( i ) . ( iii ) A longer but more direct proof is as follows ...
Table des matières
PART | 1 |
VECTOR SPACES AND LINEAR MANIFOLDS | 39 |
THE ALGEBRA OF MATRICES | 72 |
12 autres sections non affichées
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Expressions et termes fréquents
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 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 V₁ values vector space view of Theorem w₁ write x₁ xTAx y₁ zero