063) > prove the following: Let To To be two topologies on X70 and denote CI,...
10. Let T1 and T2 be two topologies on a set X. Then T1 is said to be a finer topology than T2 (and T2 is said to be a coarser topology than T1) if Ti2 T2. Prove that (i) the Euclidean topology R is finer than the finite-closed topology on R; (ii) the identity function f: (X, Ti) -(X, T2) is continuous if and only if TI is a finer topology than T2.
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Problem II. i) Let Tı and T2 be two topologies on the same space X. Suppose that T2 is finer than η. If (X,n) is compact, does it follow that (X,2) is compact? Conversely, if (X, T2) is compact, does it follow that (X, Ti) is compact? la. ii) Let Y C X be equipped with the subspace topology. Show that Y is compact in the subspace topology if and only if any cover of Y with open sets in...
(a) On R2, prove that di ((zı, y), (z2W2)) := Izı-zal + ly,-Val is a metric. (b) Assume that doc ( (zi, yī), (z2,p)) := maxlz-zal, lyi-yl} is a metric on R2 for each p 21. Prove that di and d induce the same topology on R2. You may use the following lemma (but do not need to prove it): Lemma: Let d and d' be two metrics on aset X; let T and T' be the topologies the induce...
3. Let X be a random variable and denote by Mx(t) its MGF. Prove that, for any t > 0, we have P[X >Mx(t)e
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the CDF is 0 x<0 0<x<2 1 2x Use the CDF to obtain the median checkout duration ù.
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A). 44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
10. Prove the following theorem Theorem 1 Let H and H denote the input-output transfer functions for the continuous time systems associated with state matrices (A, B, C) and (A, B,C), respectively. Thus the systems have state representations (t) = Ar(t)+Bu(t) t)C(t) 1(t y(t) and Ci(t) = Assume system (A B. C) and (A. B,C) are equivalent representations, and hence there erists an invertible matriz P such that i(t) = Pa (t) defines a coordinate transformation between the two systems...
Let 4. ) Using only the definition of infinite Series convergence, prove the following: w, ZER. Given of in and on respectively convergent to X and Y, then zyn =wX t zY In are 2 WXN t DE 6 Use the theorem above to prove the following: Let WEIR. Given to and are respectively convergent to X and Y, then £ w x n = wX,
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the CDF is x<0 x2 FX) OSX<2 25x Use the CDF to obtain the median checkout duration M.