# Homework Help: Linear Transformation Question

1. Mar 14, 2013

### HarryHumpo

1. The problem statement, all variables and given/known data

Let V = F^n
for some n ≥ 1. Show that there do not exist linear maps
S, T : V → V such that ST − T S = I.

3. The attempt at a solution

I used induction to prove that ST^n-T^nS = nT^n-1 and that S^nT-TS^n=nS^n-1, and I know I'm supposed to use that to come up with a contradiction to the fact that the space is finite dimensional, but I'm not sure how to approach that part of the problem.
Any help is appreciated

2. Mar 14, 2013

### Staff: Mentor

I don't understand why you're doing this; i.e., looking at Tn.

V is an n-dimensional vector space over some field F - that's what Fn means.

It seems to me that the most obvious way to approach this problem is to assume that there are linear maps from V to V such that ST - TS = I. If you get a contradiction, and you should, you can conclude that no such maps exist.

3. Mar 14, 2013

### HarryHumpo

The hint that was given was to prove that ST^n-T^nS = nT^n-1 (which I did) and see how it bring about a contradiction, but I don't see how the second part works.

4. Mar 14, 2013

### Dick

You know every matrix satisfies its characteristic polynomial. Try working with that. The easy alternate way to do it is to take the trace of both sides. I think both might get into trouble if the field has finite characteristic. Can it?

5. Mar 14, 2013

### HarryHumpo

Looking at the trace it seems so obvious now, thank you!

6. Mar 14, 2013

### Dick

That has some qualifications depending on what F is. In some finite fields, trace(I) is zero. You can also argue using the hint and the characteristic polynomial, but I think that has the same qualification. If the field is such that 1+1+...+1 (n times) isn't equal to zero, then it should work.