A crypto scam and nilpotency

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!

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):

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: ETH and USDT.

• Did you profit from the operation?
• What is the linear transformation underlying applying the formula? Provide its matrix .
• Is injective? Is surjective?
• What is the kernel of ? What is the image of ?
• 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 (positive) account balances could have led to profit?

Definition

A transformation is called nilpotent if . In other words, after applying a sufficient amount of times (depending on ), the resulting transformation sends all vectors to . The matrix corresponding to the transformation is also often called nilpotent where this does not create ambiguity.

One example of a nilpotent transformation is the transformation itself, given by a matrix with all entries equal to . The aim of this problem is to discover that there is a rich world of nilpotent transformations.

• Explain why all nilpotent transformations should have a non-zero kernel.
• Consider matrices of the following form: all the entries are zero except entries one above the diagonal that are equal to . Without explicitly calculating powers of , explain why is nilpotent for all . It can be a good idea to start by considering small matrix sizes ( or ).
• Denote by the linear transformation corresponding to multiplying by . For , find the dimensions of the kernels of the powers of (in symbols, find , , , …). Explain your answer.
• Consider the following matrix of a linear transformation :

What is the dimension of ? ? ? … In other words, what are the dimensions of the kernels of the powers of ?

Not used

• Choose any non-zero nilpotent matrix such that it has no zero entries (except the one provided on the Wikipedia page for nilpotent matrices as an example). Find a basis of such that