Product of Transpost of Linear Operators

Question

Let $T_1,T_2\in \text{End}(V)$, where $V$ is a finite dimensional vector space. If $T_1^*,T_2^*\in \text{End}(V^*)$ are the tranposes of $T_1$ and $T_2$ respectively, show $(T_1T_2)^*=T_2^*T_1^*$

I do apologize if this problem seems rather trivial, but I am really confused on where to start.

I know the transpose of a linear operator $T^*\in \text{End}(V^*)$, where $V^*$ is the dual space of $V$, has the property $(T^*f)(x)=f(Tx)$, for $x\in V$ and $f\in V^*$.

I am just not sure how to go about it.

Any hints/help would be much appreciated. Thanks in advance!

Answer 1

$(T_2^*T_1^*f)(x)=(T_1^*f)(T_2x)=f(T_1T_2x)$.