Arbeiten zur deskriptiven Mengenlehre (Mathematische by A.A. Ljapunow, E.A. Stschegolkow and W.J. Arsenin

By A.A. Ljapunow, E.A. Stschegolkow and W.J. Arsenin

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Extra info for Arbeiten zur deskriptiven Mengenlehre (Mathematische Forschungsberichte, 1)

Example text

Xk are orthonormal and the vectors x1 , . . , xk , xk+1 , . . , xn are linearly independent, applying the Gram–Schmidt process to the latter list produces the list x1 , . . , xk , z k+1 , . . , z n of orthonormal vectors. The Gram–Schmidt process may be applied to any finite list of vectors, independent or not. If x1 , . . , xn are linearly dependent, the Gram–Schmidt process produces a vector yk = 0 for the least value of k for which xk is a linear combination of x1 , . . , xk−1 . 5 Orthonormal bases.

The function A → adj A is continuous on Mn (each entry of adj A is a multinomial in the entries of A) and every matrix in Mn is a limit of nonsingular matrices, so properties of the adjugate can be deduced from continuity and properties of the inverse function. For example, if A, B ∈ Mn are nonsingular, then adj(AB) = (det AB)(AB)−1 = (det A)(det B)B −1 A−1 = (det B)B −1 (det A)A−1 = (adj B)(adj A). 3) For any c ∈ F and any A ∈ Mn (F), adj(c A) = cn−1 adj A. In particular, adj(cI ) = c I and adj 0 = 0.

An ∈ F, diag(a1 , . . , an ) always denotes the matrix A = [ai j ] ∈ Mn (F) such that aii = ai for each i = 1, . . , n and ai j = 0 if i = j. Suppose that D = [di j ], E = [ei j ] ∈ Mn (F) are diagonal and let A = [ai j ] ∈ Mn (F) n be given. Then (a) det D = i=1 dii ; (b) D is nonsingular if and only if all dii = 0; (c) left multiplication of A by D multiplies the rows of A by the diagonal entries of D (the ith row of D A is dii times the ith row of A); (d) right multiplication of A by D multiplies the columns of A by the diagonal entries of D, that is, the jth column of AD is d j j times the jth column of A; (e) D A = AD if and only if ai j = 0 whenever dii = d j j ; (f) if all the diagonal entries of D are distinct and D A = AD, then A is diagonal; (g) k k for any positive integer k, D k = diag(d11 , .

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