By Zhe-xian Wan.

1. Linear Algebra over department earrings --
2. Affine Geometry and Projective Geometry --
3. Geometry of oblong Matrices --
4. Geometry of other Matrices --
5. Geometry of Symmetric Matrices --
6. Geometry of Hermitian Matrices.

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Additional resources for Geometry of matrices : in memory of Professor L.K. Hua (1910-1985)

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8: Let D be any division ring, n be any integer > 1, and A be a bijective map of AGl(n, D) to itself which carries lines into lines. Then A * carries lines into lines, too. If A carries also planes into planes, then so does A-1. Proof: It is enough to prove that any line of AGl(n,D) is the image of a line of AGl(n,D). Let / be any line of AGl(n,D) and P i , P 2 be two distinct points on /. Then ^4 _1 (Pi) and *4 -1 (P 2 ) are two distinct points of AGl(n,D). There is a line passing through A~1(P\) and ^4 _ 1 (P 2 ), and call it by /'.

Affine Spaces and Affine Groups Now we shall introduce the affine group. We call the transformation of AG\n,D) to itself AG\n,D)-+AGl{n,D) (xi,a? ne transfor­ mation of the n-dimensional left affine space AG'(n, J5). Clearly, an affine transformation is a bijective map from AGl(n,D) to itself and the set of affine transformations of AG\n, D) forms a group with respect to the com­ position of maps, which is called the affine group of the n-dimensional left affine space AGl(n, D) over D and denoted by AGLn(D).

If K is an alternate matrix, then it is easy to see that txKx = 0 for all column vectors x G F^n\ Conversely, suppose that txKx = 0 for all x G F^n\ From t e{Kei = 0 we deduce ku = 0 (1 < i < n), and from *(et- + ej)K(ei + ej) = 0 we deduce kij — —kji for all i ^ j . Hence K is alternate if and only if l xKx = 0 for all x G F^n\ If F is of characteristic not two, then alternate matrices are usually called skew-symmetric matrices. Two n x n alternate matrices K\ and K2 are said to be cogredient if there is an element P G GLn(F) such that ' P / ^ P = K2.

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