An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 41
... ADDITION , i.e. addition of vectors of the same order is defined by the formula ( x1 , ... , xn ) + ( Y1 , ... , Yn ) = ( x1 + Y1 , ... , xn + Yn ) . From the last two definitions a number of important con- sequences can be inferred ...
... ADDITION , i.e. addition of vectors of the same order is defined by the formula ( x1 , ... , xn ) + ( Y1 , ... , Yn ) = ( x1 + Y1 , ... , xn + Yn ) . From the last two definitions a number of important con- sequences can be inferred ...
Page 72
... ( addition and multiplication by scalars ) to which vectors could be subjected , we proceeded to study the formal nature of these operations . A similar programme is to be carried out in the present chapter with respect to matrices ...
... ( addition and multiplication by scalars ) to which vectors could be subjected , we proceeded to study the formal nature of these operations . A similar programme is to be carried out in the present chapter with respect to matrices ...
Page 79
... addition is commutative and associative , A + B = B + A , A + ( B + C ) = ( A + B ) + C . Furthermore , matrix addition and multiplication by scalars are connected by the following distributive laws : a ( A + B ) = αA + αB , ( a + B ) A ...
... addition is commutative and associative , A + B = B + A , A + ( B + C ) = ( A + B ) + C . Furthermore , matrix addition and multiplication by scalars are connected by the following distributive laws : a ( A + B ) = αA + αB , ( a + B ) A ...
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