An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 163
... once to Theorem 5.6.2 . - = Denote by the vector space of all vectors BCx such that ABCX 0 , and by the vector space of all vectors Bx such that ABX = 0. Clearly U c V , and so d ( U ) ≤ d ( V ) . But , by Theorem 5.6.4 , d ( u ) = R ...
... once to Theorem 5.6.2 . - = Denote by the vector space of all vectors BCx such that ABCX 0 , and by the vector space of all vectors Bx such that ABX = 0. Clearly U c V , and so d ( U ) ≤ d ( V ) . But , by Theorem 5.6.4 , d ( u ) = R ...
Page 203
... once again . Among all non - zero polynomials annihilating A we now consider those of least degree , and by multiplying them by suitable non - zero constants we ensure that they are monic ( i.e. they have their leading coefficients ...
... once again . Among all non - zero polynomials annihilating A we now consider those of least degree , and by multiplying them by suitable non - zero constants we ensure that they are monic ( i.e. they have their leading coefficients ...
Page 307
... once that it continues to hold for the latter . The argument is by induction with respect to n and is of the type with which we are now familiar from the proofs of Theorems 10.3.4 , 10.3.7 , and 10.3.8 . Let us assume that the assertion ...
... once that it continues to hold for the latter . The argument is by induction with respect to n and is of the type with which we are now familiar from the proofs of Theorems 10.3.4 , 10.3.7 , and 10.3.8 . Let us assume that the assertion ...
Table des matières
PART | 1 |
VECTOR SPACES AND LINEAR MANIFOLDS | 39 |
THE ALGEBRA OF MATRICES | 72 |
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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 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