An Introduction to Linear AlgebraClarendon Press, 1955 - 433 pages |
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Page 54
... suppose α1 X1 + ... + αk Xk + αk + 1Xk + 1 = 0 . that Then α + 1 = k 0 , for otherwise we should have X + 1 Є N. Hence α 1 X1 + ... + αx Xx = 0 , 1 and , since X1 , ... , X are linearly independent , it follows that a0 . The elements X1 ...
... suppose α1 X1 + ... + αk Xk + αk + 1Xk + 1 = 0 . that Then α + 1 = k 0 , for otherwise we should have X + 1 Є N. Hence α 1 X1 + ... + αx Xx = 0 , 1 and , since X1 , ... , X are linearly independent , it follows that a0 . The elements X1 ...
Page 169
... suppose that ( 6.1.1 ) holds for every x Є S and for every = 0 ; in other words , suppose that ƒ is invariant under every operator of the set . If now the determination of f ( x ) , for some particular xo e S , is difficult or tedious ...
... suppose that ( 6.1.1 ) holds for every x Є S and for every = 0 ; in other words , suppose that ƒ is invariant under every operator of the set . If now the determination of f ( x ) , for some particular xo e S , is difficult or tedious ...
Page 218
... Suppose that all characteristic roots of I - A are less than 1 in modulus . Prove that 0 < | det A ] < 2 " ; and show that this result is best possible . 27. Let A = ( a ) be a real nxn matrix and suppose that Arr > Σlars ( ~ = 1 ...
... Suppose that all characteristic roots of I - A are less than 1 in modulus . Prove that 0 < | det A ] < 2 " ; and show that this result is best possible . 27. Let A = ( a ) be a real nxn matrix and suppose that Arr > Σlars ( ~ = 1 ...
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