By Spencer J. Bloch, R. Keith Dennis, Eric M. Friedlander, Micahel Stein (ed.)

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**Sample text**

17: Any m x n matrix of row rank r over a division ring is equivalent to I(r) r\(m—r,n—r) Proof: Let A be an m x n matrix of row rank r over a division ring D. An mxm matrix P is called a permutation matrix if every row and every column of P has only one 1 and m — 1 O's. Clearly lPP — / , hence permutation matrices are invertible matrices. Moreover, PA is a matrix obtained from A by permuting its rows. Since A is of row rank r, we may choose an mxm permutation matrix P such that the first r row of PA are linearly independent, then the last m — r rows of PA are linear combinations of the first r rows.

6 Comments In order to be as self-contained as possible the book starts with a chapter on linear algebra over division rings. The material of this chapter is rather standard and can be found in many books. We follow mainly Hua and Wan 1963 and Wan 1992b. 1 Affine Spaces and Affine Groups Let D^ be the n-dimensional row vector space over D. The vectors in D^n\ both the non-zero vectors and the zero vector, will now be called points; the cosets of D^ relative to any 1-dimensional subspace, lines; the cosets of Z)(n) relative to any 2-dimensional subspaces, planes; and the cosets of D^ relative to any (n — l)-dimensional subspace, hyperplanes.

15 there is an (n — r) x r matrix D such that (' R R\ Q= [») is an n x n invertible matrix. 18: The row rank and column rank of any m x n matrix over a division ring are equal. 10: Let A be an m x n matrix over a division ring. Then the common value of the row rank and column rank of A is called the rank of A, which is denoted by rank A. 2. 19: Let A and B be m x n matrices over D. Then A and B are equivalent if and only if they have the same rank. 20: Let A be an m x n matrix of rank r over a division ring D.