By Vitushkin A. G.

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**Example text**

Now if x, E S. finition. x. E S, and X, e S2; also since "2 e S, then X2 e S. and Xl e S2; but smce S, and S2 are vector spaces, it follows that OIX,+OZx2eS, and for all scalars 0, and 0z; and hence a. +o2x2eS2 + Of Xl e S. This completes the proof. I Mathematical induction can be used to show that if S = S n S n'" rv S for any positive integer m, then S is a vector subspace of E. if each ~, is a1vector subspace of En. 1 h~ the ~'ollowing geometric interpretation: Let be a plane through the ongm and of dimension k in E" (the points of ~k' when viewed as vectors, form a subspace 5'.

A; A E R}. But, by Def. 2, this line is parallel to the line f£* that passes through the origin and the point b - a, where f£" is defined by fR" = [x : x = (b - a)1 + (I - },)O; ),ER}, which reduces to Two lilies ill Ell are parallel vectors. 8. We also define two lines in E. to be perpendicular they have orthogonal direction vectors. ; ). E R}. But a direction vector of the line ~* is b - a and this uniquely determines the direction of f£*, and since f£* passes through the origin, :t'* is uniquely determined.

Find the direction angles of the line through the points a' = [2,0, -I] and b' = [I, -1,3], 8. Find the distance from the point x' = [2, 1, -IJ to the line 5£ that goes through the two points a'". [I, 1,0] and b' = [I, -1,2]. 15. Let fil be a line through 0 and a, where a' = [I, -1, 1]. J that is perpendicular to ~ and passes through the point [2, 1, -1]. 16. Find the projection of the directed line segment ( from a, to a1 onto the line fil that goes through 0 and b where al = [1,2, -1], a2 ...