A bayesian justification for the linear pooling of opinions by Bacco M., Mocellin V.

By Bacco M., Mocellin V.

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Show that A, B, C are not colinear iff {AB, AC} is a set of linearly independent vectors. 19. Let S = {u, v, w}. 'F(S) defined by f(u) = 1 f(v) = I f(w) = 1 g(u) = 1 g(v) = 1 g(w) = 0 h(u) = 1 h(v) = 0 h(w) = 0 are a basis for fF(S). Find the coordinates of the characteristic functions Xu, Xv, Xw relative to this basis. 53 7 The elements of vector spaces: a summIng up Our objective in this section is to work out a number of numerical examples to illustrate and illuminate the theory of vector spaces we have developed so far.

9. The set of all continuous functions y = f(x), ferential equation 00 < x < 00 satisfying the dif- y" - y' - 2y = 0 is a vector space. ) X 31 4: Examples of vector spaces 10. x)y(n) + a1(x)y(a, b) = {Polynomials of x, a ::; x ::; b}. c1>(a, h) c CiJ c ceCa, h).

13. Let g; be the subspace of 1R3 given by g; = {(x, y, z)ly - z = O}. 07 = 1R3. 14. Under what conditions on the number a will the vectors (a, 1,0), (1, a, 1), (0, 1, a) be a basis for 1R3? 15. Let AI, ... , An be vectors in "1'. Suppose that n = dim "1'. Show that {AI' ... , An} is linearly independent iff dim ~(AI' ... , An) = n. 16. The equation y = 3x defines a straight line in the xy-plane. Show that if A, B are on this line then A, Bare linearly dependent vectors. 17. Let A, B, C, D be four distinct points on a plane AD form a set of linearly dependent vectors.

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