By Karlheinz Spindler

A complete presentation of summary algebra and an in-depth therapy of the purposes of algebraic options and the connection of algebra to different disciplines, resembling quantity conception, combinatorics, geometry, topology, differential equations, and Markov chains.

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F or points p]^, . . , p ^ on S. and . . , <3^ on we denote O P l ---Pm A qi--- V if TTq (pj) = qj for I < j < m . (See F ig. 1 . ) This is also a cla ssical notation. 17 (Desargues). Let Apqr and Ap’ q’ r* be triangles in IP^. 16 the lines pp^, qq^, and rr* meet at one point if and only if the points a = pq П P^q*, b = qr П q^r^ and c = rp П r ’p* are collinear. (See Fig. ) P roof: Suppose that pp*, qq^ and rr^ meet at a point o . Put s = ac П op, t = ac П "oq, U = ac П or, and u^ = BF П o r.

We leave the proof of the following theorem to the reader (Exercise 2). 14 (Steiner). Let p and p ’ be distinct points on an irreducible conic C. Take lines i and i' such that p ¿ i and p ’ ¿ i*. 41 Ф = 7Г ,7 Г -Ч jg — i* P' P is a projective transformation. Conversely, if points p, p* (p p*), lines j2’ (P ^ j2, P* ^ j2’ ) and a projective transformation ф: j2 — are given, then the closure o f the set { q = рт П р*ф(г) I r e f } is either an irreducible conic passing through p and p ’ o r a line.

If this is the case, I is said to be tangent to C al (0: X^: Х з). (See Fig. ) The proof o f the following lemma is left to the reader (Exercise 2) . 2 . Let C be an irreducible con ic. (I) If p e C, then C has a unique tangent line to C at p (which is denoted by TpC). (2) If p e IP^ - C, then exactly two lines pass through p which are tangent to C. (See Fig. 3 . Any two irreducible conics can be mapped each other by projective transform ations. P roof. Let C be an irreducible con ic. It suffices to show that the equation of C is given by XqХз - X j = 0 in a suitable homogeneous coordinate system (Xq : Xi : X 3 ) .