Is T^n Linear When T is Linear?

[cristina89]

Homework Statement

If T is a linear transformation, proof that Tⁿ is a linear transformation (with nEN).

Homework Equations

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

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.

 

[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

 

[HallsofIvy]

You can do the general proof "by induction".

 

[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:

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

 

[HallsofIvy]

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

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