By Frithjof Nevanlinna, Rolf Nevanlinna
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Linear Algebra Remark 2. It follows from the above that A Rs and KV(A *) are linearly independent complements in Ry and that the same is true for A* Re,. (A). 2. The results of exercise I contain the complete theory of systems of real linear equations. Verify the following main theorems: a. ,n) ZajE'=O i-1 is of rank r, the system has precisely sit - r linearly independent solution vectors x a,. i-s b. ,m) the equation holds. Then if xQ JO' ai is a particular solution of the nonhomoge- neous system, one obtains the general solution by adding the general solution of the corresponding homogeneous system of equations.
Must be an odd permutation of the ordering ... ,...... T", xkO, of the vertices of s"h. But then k R R , xp-t, x, xp+t, . xo, . k is likewise an odd permutation of the ordering k k k and the corresponding J-terms in the above double suns consequently cancel each other out. Thus in this double sum only those terms remain which come from side simplexes s'"-' of the subsimplexes that contain no interior and thus only boundary points of the decomposed simplex s"'. If in addition the point x is now shifted to a vertex of the simplex, to x0, for example, then only the terms which correspond to the side simplex s,'-' of s" opposite to xo remain.
One has. then found in turn to - r positive or negative pairwise orthogonal unit vectors which are completed, with an arbitrary coordinate system for the null space R' orthogonal to these unit vectors, to a complete coordinate system which is orthonornial with respect to B. If p g 0 of these vectors are positive and q > 0 are negative, then p + q + r = tit; the positive unit vectors generate a p-dimensional subspace U. where B is positive definite ; in the q-dimensional subspace generated by the negative unit vectors B is negative definite; and It" = R' contains nothing but null vectors.