An Introduction to Linear AlgebraClarendon Press, 1972 - 440 pages |
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Page 148
... once led to the general solution in the form ( 5.3.7 ) . In § 6.3.2 we shall see how the method for solving systems of linear equations illustrated above can be used with even less labour . 5.4 . Systems of homogeneous linear equations ...
... once led to the general solution in the form ( 5.3.7 ) . In § 6.3.2 we shall see how the method for solving systems of linear equations illustrated above can be used with even less labour . 5.4 . Systems of homogeneous linear equations ...
Page 163
... once to Theorem 5.6.2 . - Denote by the vector space of all vectors BCx such that ABCX 0 , and by V the vector space of all vectors Bx such that ABX = 0. Clearly Uc 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 V the vector space of all vectors Bx such that ABX = 0. Clearly Uc 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 ...
Table des matières
DETERMINANTS VECTORS MATRICES | 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 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