Let A e Cpxp,A e C, and let 11-11 a multiplicative norm on Cpxp. Use Theorem 7.27 of your lecture notes to show that if Al > ll All , then 1. XIp A is invertible and Ip Theorem 7.27. If Il is a su...
The following Theorem states: If II. Il is a sub-multiplicative norm on Cp*P, then | 2 1. Moreover, if X E C and |XIl<1, then Ip X is invertible . (lp-X)-1 = 0X1; i.e., the sum converges b-X)-11s Prove the following: Let A E Cpxp,A E C, and let II . Il be a multiplicative norm on Cpxp. Show that if Ιλ/> 11 A 11, then 1)2lp A is invertible and 1시-11시
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli 5. Let Zli_ {a + bi l a,b...
Proposition 7.27. Suppose fn: G + C is continuous, for n > 1, (fn) converges uniformly to f :G+C, and y C G is a piecewise smooth path. Then lim n-00 $. fn = $. . 7.23. Let fn(x) = n2x e-nx. (a) Show that limn400 fn(x) = 0 for all x > 0. (Hint: Treat x = O as a special case; for x > 0 you can use L'Hôspital's rule (Theorem A.11) — but remember that n is...
part (c) 7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn be an (n x n) matrix and be R. Consider the problem 1 (P2) min2+ s.t. xe R" 1Ax-bil2 1 where & > O is fixed and Il IIl denot es the 2-norm. Call g.(x)=l|2 the objective function of problem (P2) 1Ax-bl2 i) [3 marks] Compute the gradient of g, and use it to show that the solution xi of this problem verifies (I+EATA)(x)...
s h) for all z c l e Sub-problem 3. Recall monotonicity of integration: If h() S [-1, 1], then This just says that integruls preserve inequalities 1. Explain why this is true graphically 2·Let g be continuous on [0,1]. Use the previous item, and the fact that to show that 3. Use the first two items to show that if g is bounded, say Ig(r)l s M for z [0, 1], then first two derivatives are continos on is...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
Please show the solutions for all 4 parts! Problem 1 Let m E Z that is not the square of an integer (ie. mメ0, 1.4.9, ). Let α-Vm (so you have a失Q as mentioned above) (i) Prove the following:Qla aba: a,b Q is a subring of C, Za]a +ba: a, b E Z is a subring of Qla], and the fraction field of Z[a] is Q[a]. (3pts) (ii) Prove that Z[x]/(X2-m) Z[a] and Qx/(x2 mQ[a]. (3pts) i Let n be...
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...