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PARALLELISM 31 hyperplane πχ have the equationy 1 ot l + · · · + ynocn = a and the hyperplane π 2 have the equation yißl + · · · + yn ßn = b. Prove that π χ || π 2 if and only if there is a y φ 0 in k such that ßt = yoLt for / = 1, . . , n. 7. Assume that a coordinate system has been chosen for the threedimensional space X. If the plane π has the equation 0 Ί ) 3 - 7 2 + 0 > 3 ) 2 + 6 = 0, find an equation for the plane which passes through the point x = (1, — 2, 7) and is parallel to π. 1 so that we can also speak about parallel subspaces of «-spaces which do not have the same dimension.

Ad_t which can be done since d < n + 1 implies that d — 1 < n. If x is a point in X, we claim that x, Atx9 . . , Ad_lx are d independent points. 1 case for d = 3. Suppose not; that is, suppose the dpoints lie in S(x9 U) where dim U = d — 2. Then the d — 1 vectors Ax = x, Axx9 . . , Ad_ x = x9 Ad_1x belong to U. But it is impossible for the (d — 2)-dimensional linear subspace U to contain the d — 1 linearly independent vectors Au . . , Ad_1. Done. 2. Let I

A dilation is completely determined by its dilation ratio and the image of one point. Proof. Let D be a dilation with ratio r. Assume that x e Jfand that the image D(x) is known. If y Φ x belongs to X, we must show that D(y) is known. 1 12. THE RATIO OF A DILATION 51 put / = x v y and let m denote the line through D(x) which is parallel to /. Since D is a dilation, D(y) e m. y)D(x) for some sek. But this is equivalent to D(x), D(y) = s(x, y); consequently, by definition of dilation ratio, s = r.