# Linear transformation proof

1. Aug 15, 2013

### cristina89

1. The problem statement, all variables and given/known data
If T is a linear transformation, proof that Tn is a linear transformation (with nEN).

2. Relevant equations
I know that T is a linear application if:
T(u+v) = T(u) + T(v)
T(au) = aT(u)

3. The attempt at a solution
Actually I don't know how to start using these two affirmations. Can anyone help me with it?
I know how to do this when it has numbers, but then it comes to these kind of proofs, I don't know how to do this.

2. Aug 15, 2013

### ehild

Start with T2. Is it true that T2(u+v)=T2(u)+T2(v)? Note that T2(u+v) means T(T(u+v)).

ehild

3. Aug 15, 2013

### HallsofIvy

Staff Emeritus
You can do the general proof "by induction".

4. Aug 20, 2013

### cristina89

I'm trying to solve it by induction.

For n = 1 ok.

Assuming that's ok for n = k.

For n = k+1

I don't know if I'm doing it right in this part:

Tk+1 = Tk.T(u+v) = Tk.(T(u+v)) = Tk(T(u)) + Tk(T(v)). Can I just afirm that's ok since T(u+v) is an application and Tk is an application too?

5. Aug 20, 2013

### HallsofIvy

Staff Emeritus
I would have put in one more step. Tk(T(u+ v))= Tk(T(u)+ T(v)), using the "given" fact that T is linear, and then "= Tk(T(u))+ Tk(T(v))" using the "induction hypothesis" that Tk is linear.

And, of course, you now need to prove that Tn(au)= aTn(u) but that can be done the same way.