An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 92
... possesses the unique solution X = A - 1B . ( 3.6.4 ) ( 3.6.5 ) ( ii ) If C is a non - singular n × n matrix and D is an m × n matrix , then the matrix equation YC = D possesses the unique solution Y = DC - 1 . It is sufficient to ...
... possesses the unique solution X = A - 1B . ( 3.6.4 ) ( 3.6.5 ) ( ii ) If C is a non - singular n × n matrix and D is an m × n matrix , then the matrix equation YC = D possesses the unique solution Y = DC - 1 . It is sufficient to ...
Page 132
... possesses no solution ; the system x1 + x2 = 2 , x1 + x2 = 0 X1 - X2 = 0 possesses precisely one solution ( x1 = x2 = 1 ) ; and the system x1 + x2 = 1 , 2x1 + 2x2 = 2 possesses an infinity of solutions ( x1 = t , x2 = 1 − t , t ...
... possesses no solution ; the system x1 + x2 = 2 , x1 + x2 = 0 X1 - X2 = 0 possesses precisely one solution ( x1 = x2 = 1 ) ; and the system x1 + x2 = 1 , 2x1 + 2x2 = 2 possesses an infinity of solutions ( x1 = t , x2 = 1 − t , t ...
Page 145
... possesses a unique solution if and only if R ( A ) = R ( B ) = n . It possesses no solution if and only if R ( A ) < R ( B ) . EXERCISE 5.3.2 . Illustrate the above result by reference to ( i ) the systems of equations mentioned ...
... possesses a unique solution if and only if R ( A ) = R ( B ) = n . It possesses no solution if and only if R ( A ) < R ( B ) . EXERCISE 5.3.2 . Illustrate the above result by reference to ( i ) the systems of equations mentioned ...
Table des matières
PART | 1 |
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 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