Given:
$V$ is a vector space.
$f: V \to V$ is a linear transformation
The pullback on $f$ on $T^{k}(V)$ is the map:
$f^{*}: T^{k}(V) \to T^{k}(V)$
$f^{*}(T)(\vec{v_{1}}, ...,\vec{v_{k}})= T(f\vec{v_{1}}, ..., f\vec{v_{k}})$, for any $T\in T^{k}(V)$
I must show that $f^{*}: T^{k}(V) \to T^{k}(V)$ is a linear transformation.
This seems to be quite straight forward, but I would just like to double check if I'm going about it correctly. Is the following what I have to show:
$f^{*}(S+T)(\vec{v_{1}}, ...,\vec{v_{k}})=(f^{*}(S)+f^{*}(T))(\vec{v_{1}}, ...,\vec{v_{k}}) $, for $S, T\in T^{k}(V)$ and
$f^{*}(aT)(\vec{v_{1}}, ...,\vec{v_{k}})=a(f^{*}(T))(\vec{v_{1}}, ...,\vec{v_{k}}) $
I'd appreciate any help!