Proving $T^{n} \in \mathcal{L}(V, V), \forall n \in \mathbb{N}$.

Question

Let $T \in \mathcal{L}(V,V)$. Then we define the map $T^{n}$ for $n \in \mathbb{N}$ recursively by

$$T^{1} = T$$ $$T^{k+1} = T \circ T^{k}$$

for $k \in \mathbb{N}$.

How would I go about proving that $T^{n} \in \mathcal{L}(V, V), \forall n \in \mathbb{N}$ using induction?

I know that from the given definition of the map $T^{n}$ that $T^{i} = T \circ T^{i - 1}$. But I am not sure where/how exactly to start the proof.

Answer 1

The map $T^1$ is simply $T$, which is linear.

And, is $T^n$ is linear, then $T^{n+1}$ is linear too, because it is the composition of two linear maps (it is equal to $T\circ T^n$).