MAT188-WRITTEN-HOMEWORK 3, Nov 6th, 11:59 PM Problem 1. You receive the following message on Telegram from the user 'thetrueelonmusk'. thetrueelonmusk: Dear user, I am happy to share that I finally managed to discover a loophole in the world's financial system: a formula that allows to generate infinite money! I am eager to share this with you as a token of gratitude for your support. To make using the formula easier, I created the website 'absolutelynotsuspicious.tv' that will allow you to use my infinite money glitch. You just need to connect you crypto account to the website, add some funds and let the formula do the work for you! 2 You decide to follow the link (on a virtual machine, of course), and, after inspecting the source code of the webpage, you uncover the formula so carefully hidden by thetrueelonmusk. It is an exchange of cryptocurrency with the following rate (USDT, ETH and TON are cryptocurrency names, and the problem does not require to know their monetary value): 1 USDT 3 1 ETH 3 1 TON 3 2 USDT+5 ETH +1 TON 2 USDT +1 ETH +5 TON -2 USDT 3 ETH - 3 TON The formula seems to work in the following way: it takes all the USDT, ETH and TON that you have on your account, and converts them all at once using the exchange. The website code then automatically applies the formula three total times (each time, to the new balance after the previous application) to maximise your gains. You seem to discover the scam: having any TON would just result in a net loss in the exchange. You therefore decide to launch the website code for an account with your ETH and USDT savings: 1 ETH and 10 USDT. (1) Did you profit from the operation? (2) What is the linear transformation underlying applying the formula? Provide its matrix A. (3) Is A injective? Is A surjective? (4) What is the kernel of A? What is the image of A? (5) What is the linear transformation underlying applying the formula twice? three times? What do you observe? In context of the problem, and assuming that the code ignores that balance cannot be negative while applying the formula repeatedly, which initial (non-negative in each currency) account balances could have led to profit? MAT188-WRITTEN-HOMEWORK 3, Nov 6th, 11:59 PM 3 Definition 1. A transformation N: R R is called nilpotent if No... o N = 0. Here 0: R" →R" denotes the zero map that sends all vectors to Ō. In other words, after applying N a sufficient amount of times (depending on N), the resulting transformation sends all vectors to ō. The matrix corresponding to a nilpotent linear transformation is also often called nilpotent where this does not create ambiguity. Definition 2. For a linear transformation T : R" → R", Tk denotes T composed with itself k times. That is Tk=To To...oT. Tk is also often called the k-th power of T. k times One example of a nilpotent transformation is the 0 transformation itself, given by a matrix with all entries equal to 0. The aim of this problem is to discover that there is a rich world of nilpotent transformations. (a) Explain why all nilpotent transformations should have a non-zero kernel. (b) Consider nxn matrices Jn of the following form: all the entries are zero except entries one above the diagonal that are equal to 1. Without explicitly calculating powers of Jn, explain why Jn is nilpotent for all n. It can be a good idea to start by considering small matrix sizes (n = 2 or 3). (c) Denote by T, the linear transformation corresponding to multiplying by Jn. For n = 5, find the dimensions of the kernels of the powers of Ts (in symbols, find dim ker T5, dim ker T3, dim ker T3, ...). Explain your answer. (d) Consider the following 5 x 5 matrix K of a linear transformation TK: R5 → R³: K = [0 1 0 0 0] 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 What is the dimension of ker Tk? dim ker T? dim ker T? dimensions of the kernels of the powers of Tk? In other words, what are the (e) Any polynomial P(x) = azª +ª‚z²+azz² +αsz+αo of degree 4 is defined by its coefficients, 5 real numbers a,..., ao. Associate to every P(x) the vector of its coefficients. Then the derivative operation can be considered as a transformation D: R5 R5 where the coefficients of the polynomial are sent to the coefficients of the derivative. (i) Show that D (derivation) is a liner transformation. You can use any results from MAT186. (ii) Find the standard matrix of D. (iii) Is D nilpotent? why or why not? (iv) Describe the kernel of D² in two ways: as span of vectors and in terms of a set of the corresponding polynomials.