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Extra resources for Chern - A Great Geometer of the Twentieth Century

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If S' is a hyperplane and dim S > 1, then S \\ S' if and only if S a S' oxS n S' = 0 . 11. Let S = S(x, U) and S' = S(y, W) be affine subspaces of X. Prove that S a S' if and only if these spaces have a point in common and U c W. ) 12. Suppose that S, S\ and S" are affine subspaces of X where S \\ S' and dim S < dim S'. If S" cz £, prove that S" || 5". Show by means of a picture that the condition dim S < dim S' cannot be omitted. 1 10. AFFINE SUBSPACES SPANNED BY POINTS 33 13. Let S be a ^/-dimensional subspace of X and x a point of X.

A. Prove that the intersection of two nonparallel traces of D is a fixed point of D. b. Show that D = lx if and only if every line in X is a trace of D. c. Assume c is the fixed point of a dilation D Φ \ x . Prove that each line through c is a trace of D. Show also that no other line could be a trace of D. d. If D is a translation TA, A φ 0, prove that / is a trace of D if and only if / has U = {A} as direction space. 17. Let k be the field Z 2 , that is, the integers modulo two. ) Prove that each dilation of (X, V, k) is a translation.

If S n 5 ' = 0 , dim(Sv S') = 1 + dim(£/ + ^ ) . d. If a point x ^ S7 dim(5 v i ) = l + dim S. e. If xl9 . . , xd are points of X, dimf^ v ··· vxd) < d — 1. 11. THE GROUP OF DILATIONS It is usually worthwhile to investigate the automorphism of a mathe­ matical structure, that is, those one-to-one mappings of the structure onto it­ self which preserve its structural properties. What are the automorphisms of affine ^-space (X, K, k) ? Affine geometry is the geometry of parallelism and it enables us to com­ pare the lengths of parallel line segments.

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