An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 165
... Show that , if m > 3 , then x1 , ... , xm are linearly dependent . = 1 , ... , n ) . 17. Show that , given any system of m linear homogeneous equations in m + 2 unknowns x1 , ... , x + 2 , there are two indices i , j in the range 1 , 2 ...
... Show that , if m > 3 , then x1 , ... , xm are linearly dependent . = 1 , ... , n ) . 17. Show that , given any system of m linear homogeneous equations in m + 2 unknowns x1 , ... , x + 2 , there are two indices i , j in the range 1 , 2 ...
Page 287
... Show that the set of all automorphisms of G is a group г. Show further that , when G is a Klein four - group , I is isomorphic to the symmetric group S3 . 6. Let G be a group and x a fixed element of G. The transformation which maps a Є ...
... Show that the set of all automorphisms of G is a group г. Show further that , when G is a Klein four - group , I is isomorphic to the symmetric group S3 . 6. Let G be a group and x a fixed element of G. The transformation which maps a Є ...
Page 422
... Show that , under the same conditions as in Theorem AXB a ... ann B. 13.5.6 , B│ > EXERCISE 13.5.5 . Show that if A and B are positive definite , then so is AxB . Schur sharpened Theorem 13.5.6 by showing that , under the same ...
... Show that , under the same conditions as in Theorem AXB a ... ann B. 13.5.6 , B│ > EXERCISE 13.5.5 . Show that if A and B are positive definite , then so is AxB . Schur sharpened Theorem 13.5.6 by showing that , under the same ...
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