By George A. F. Seber

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63. Given any set A E R", the set CA of points generated by taking the convex combination of every finite set of points xi in A, namely alxl+ a2x2 + . . + akxk (each cyi 2 o and C ai = 1) 2 is a convex set containing A. The set CA is the smallest convex set containing A and is called the convex hull of A. It is also the intersection of all convex sets containing A. 29. Given A a set in R", we define x to be an inner (interior) point of A if there is an open sphere with center x that is a subset of A; that is, there exists 6 > 0 such that Sg = {y : y E R", (y - x)'(Y - X) < 6 ) C A.

41. Rao and Rao [1998: 240-2411. Multiply the first equation by P to prove - P) = 0. 42a. C ( P ) Pvl = v1, and V1 as P projects onto V1. Conversely, if v1 E V1, then C C(P); (b) is similar. 43. Meyer [2000a: 6341. 21. Suppose U has an inner product (,), and let V be a vector subspace with orthogonal complement V I , namely V' Then U E v}. V' so that every v E U can be expressed uniquely in the form The vectors v1 and v2 are called the orthogoad projections of v onto V and V', respectively (we shall omit the words "along V'" and "along V" , respectively).

9. Halmos [1958]. 10. Rao and Bhimasankaram [2000: 391. 11a-b. Proofs are straightforward. 2 . Meyer [2000a: 2051 and Rao and Bhimasankaram [2000: 481. 12. Firstly, aj = birij so that A = BR. 12), which is a contradiction. 11. Let V1 and Vz be two vector spaces over the same field 3. Then a map (function) from V1 to Vz is said to be an isomorphism if the following hold. , + is one-to-one and onto). (2) + ( u + v ) = + ( u ) + + ( v ) f o r a l l u , v E V l . (3) +(au)= Q+(u)for all V1 cy E F and u E V1.