Let s = {k=1CkXAz be a simple function, where {A1, A2, ... , An} are disjoint. Prove that for every p>0, |CK|PXAR
b. Let U2 u~xã. Show that E(b)=n-2 for n > 2. LU
Let G = (V, S, R, S) be a grammar with V = {Q, R, T}; { = {q, r,ts}; and the set of rules: SQ Q→ RqT RrrT QQr T>t | StT b. (15) Convert G to Chomsky normal form.
2. Consider the Fibonacci sequence {rn} given by x1 = 1, 22 = 1 and Xn = In-1 + In-2 for n > 3. Using Principle of Mathematical Induction show that for any n >1, *-=[(4725)* =(4,799"]
5. ??? v. (0) = 20Y ?? v, v * ??? i ?????? t> 0 5. Let v(0) 20V. Find vov and i, for t>o 8? 5? 20.5FC Figure.5
5. Let A € Mnxn(C) with characteristic polynomial p(x) = cxºII-1(d; – x) and li + 0, Vi, a E Z>o. Show that if dim(ker(A))+k=n, then A= C2 for some complex matrix C.
1. Let a=(ay, ay) and b= (6,62) be vectors in Rº. a. Verify that <a, b >= 20,62 +5 a,b2 satisfies the inner product axioms. b. a=(-1,3), b= (2,5) Find da,b).
v Problem 5 Let Xi, і ї, , n, n-256, be i.i.d. Pois(1)-random variables, and Sn- il Xi. a) Using Chebychev's inequality, estimate the probability that P(Sn > 2E S]).
Problem #3: Let R4 have the inner product <u, v> = ulv1 + 2u2v2 + 3u3v3 + 40404 (a) Let w = (0,9,5,-2). Find llwll. (b) Let W be the subspace spanned by the vectors U1 = = (0,0, 2, 1), and u2 = (-3,0,–2, 1). Use the Gram-Schmidt process to transform the basis {uj, u2} into an orthonormal basis {V1, V2}. Enter the components of the vector v2 into the answer box below, separated with commas.
Q 3 a) Let n > 2 be an integer. Prove that the set {z ET:z” = 1} is a subgroup of (T, *). Show that it is isomorphic to (Zn, + mod n). b) Show that Z2 x Z2 is not isomorphic to Z4. c) Show that Z2 x Z3 is isomorphic to 26.