Geometric Algebra (Dover Books on Mathematics) by Emil Artin

Author:Emil Artin [Artin, Emil]
Language: eng
Format: epub, azw3
Publisher: Dover Publications
Published: 2016-01-19T18:30:00+00:00

is a trace-preserving homomorphism of T′, hence a certain element of k′ which we may denote by .

c) The map given by is an isomorphism of k onto k′.

d) If coordinates are given in Π then the coordinate system of Π′ can be placed in such a position that the image of the point (x, y) in Π is the point of Π′.

e) Describe all such maps λ by formulas.

8) Using Figure 6 give a sequence of three projections which thereby produce a projectivity σ of the line l through A, B onto itself with the following properties: A and B are fixed points; if C is distinct from A and B then its image under σ is the fourth harmonic point D. What is the order of the projectivity σ?

9) Let k be a commutative field of characteristic ≠ 2, V a vector space over k and the corresponding projective space. Let σ be a projectivity of onto itself which is of order 2 and has at least one fixed “point”. Prove that among the linear maps of V onto itself which induce σ there is one—call it λ—which is also of order 2. Show that V is the direct sum of two subspaces U and W such that λ keeps every vector of U fixed and reverses every vector of W. The projectivity σ is completely described by the pair U, W of subspaces (U, W equivalent to W, U).

10) Let k be commutative and of characteristic ≠ 2. Show that a projectivity of a line l which is of order 2 and which has a fixed point is by necessity a map of the type described in 8). Avoid any computation. Returning to 9) describe the projectivity σ by geometric constructions. Assume that k is the field of quaternions. Give a projectivity of a line which is of order 2 and is not of this type.

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