An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 9
... rows in D = | aj | n . Then ‡ ( aj ( i #r ; i 8 ) a'ij = asi ( i = r ) ( ari ( i ( i = 8 ) . Hence , by Definition ... rows ( or two columns ) of a determinant are identical , then the determinant vanishes . Let D be a determinant with ...
... rows in D = | aj | n . Then ‡ ( aj ( i #r ; i 8 ) a'ij = asi ( i = r ) ( ari ( i ( i = 8 ) . Hence , by Definition ... rows ( or two columns ) of a determinant are identical , then the determinant vanishes . Let D be a determinant with ...
Page 137
... rows ( or columns ) of a matrix . This terminology introduces no new ideas and simply means that the rows or columns of the matrix are treated as vectors . DEFINITION 5.2.3 . ( i ) The ROW RANK ( COLUMN RANK ) of a matrix AO is the ...
... rows ( or columns ) of a matrix . This terminology introduces no new ideas and simply means that the rows or columns of the matrix are treated as vectors . DEFINITION 5.2.3 . ( i ) The ROW RANK ( COLUMN RANK ) of a matrix AO is the ...
Page 139
... rows , say 51 , ... ,,, and every row of A is expressible as a linear combination of 1 , ... , . Denote by V the vector space ( of order n ) spanned by all rows of A. Since ( in view of Exercise 2.25 , p . 48 ) V is also spanned by 1 ...
... rows , say 51 , ... ,,, and every row of A is expressible as a linear combination of 1 , ... , . Denote by V the vector space ( of order n ) spanned by all rows of A. Since ( in view of Exercise 2.25 , p . 48 ) V is also spanned by 1 ...
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