# Linear algebra methods in combinatorics with applications to by LÊ¹aszloÌ Babai By LÊ¹aszloÌ Babai By LÊ¹aszloÌ Babai

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Of course, linear maps are semi-linear with Ki = K2 = K and a = i d ^ - Another important special case are the conjugate-linear maps determined by i f = C and the conjugation a(a) = a in C . If Ki = K2 = K is a. skew field and V = W is a right vector space over K, then for every jj, G K* the dilation di_t := f e V 1-^ ^jj, e V is semi-linear with respect to the inner automorphism a = o-^-i; recall t h a t the inner automorphism a^ of a skew field K defined by z/ 7^ 0 is a,y : a E K 1-^ a,y{a) := i/ai/^ G K.

The colhnear maps with D i m / ( P " ) > 2 can be represented as the superposition of a colhneation with a (general) central projection to be introduced in the following example. Example 1. General central projections. Let Q™ C P", 0 < TO < n, be a projective subspace, and let B C P " a complementary subspace to Q™. The central projection p from P " onto Q™ with center B is defined to be the map pix):=ixVB)AQ:^, xeP:. 1 immediately implies 26 1 Projective Geometry Dim(a; \/ B) = Dim x + Dim B + 1 = Dim B + 1 = n - m as well as Dim((a; V B) A Q™) = 0, so that (3) indeed uniquely determines a point of <5™.

In analogy to Corollary 8, describe all collinear maps / : P" -^ Q™ with D i m / ( P ^ ) = 0. Exercise 7. Let / G A u t P " , n > 2, and let F be the automorphism of the lattice [^", C] determined by / according to Exercise 5. Prove that / is the identity of P " , if for some k, 0 < k < n, the restriction of F to Pn,k is the identity. Exercise 8. Coarse Classification of Collinear Maps. Let PO,QTJ projective spaces over the skew field K. Define: T:={f:P:^ n,m > 2, be Q ^ l / collinear, D i m / ( P : ) > 2}, G:= AutQ" X AutP".