An Introduction to Linear AlgebraClarendon Press, 1955 - 433 pages |
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Page 63
... Consider the case n = 3. If a system of rectangular coordinates is introduced in ordinary geometrical space S , every point P becomes associated with a definite vector OP = x with components ( x , y , z ) ; and we thus have a biunique ...
... Consider the case n = 3. If a system of rectangular coordinates is introduced in ordinary geometrical space S , every point P becomes associated with a definite vector OP = x with components ( x , y , z ) ; and we thus have a biunique ...
Page 131
... consider the simplest cases of our problem . 5.1.1 . DEFINITION 5.1.1 . A LINEAR EQUATION in the unknowns x1 , ... , xn is an equation of the form a 1 x1 + ... + αn xn = b . ( 5.1.1 ) If b = 0 , then ( 5.1.1 ) is known as a HOMOGENEOUS ...
... consider the simplest cases of our problem . 5.1.1 . DEFINITION 5.1.1 . A LINEAR EQUATION in the unknowns x1 , ... , xn is an equation of the form a 1 x1 + ... + αn xn = b . ( 5.1.1 ) If b = 0 , then ( 5.1.1 ) is known as a HOMOGENEOUS ...
Page 141
... Consider r columns of A which con- tain a critical minor , say △ , and assume , without loss of generality , that A * 1 , ... , A , are such columns . By hypothesis , △ is also a critical minor of B and therefore , by Theorem 5.2.1 ...
... Consider r columns of A which con- tain a critical minor , say △ , and assume , without loss of generality , that A * 1 , ... , A , are such columns . By hypothesis , △ is also a critical minor of B and therefore , by Theorem 5.2.1 ...
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