What is meant by the notation "Tf"? I have included a screenshot of the problem I am working on for context. T is a transformation. What is meant by Tf? Is it equivalent to T(f) or does it mean T times f?

 A: It means $T(f)$: that is, $T$ turns polynomials into matrices, and $T(f)$ is the matrix that $T$ turns $f$ into.
A: The notation $Tf$ means that $T$ is a linear operator. Linear operators are often written without parentheses. You could write $Tf=T(f)$. Note that $f$ in this context is a function and not the result of applying a function.
Linear means that for two function $f$ and $g$ and coefficients $a$ and $b$ you have
$$ T(af + bg) = aTf + bTg.$$
(Linear operators on $\mathbb R^n$ can be written as matrices. You apply matrices by matrix multiplication, thus $Tf$ can be interpreted as some kind of product.)
A: $T$ is a map from $P_2$ to $M_{2\times 2}$. $Tf$ is the map applied to $f$.
A: For the matrix, observe:
$$T(1)=\left(\begin{array}{cc}0&2\\0&0\end{array}\right)=2\left(\begin{array}{cc}0&1\\0&0\end{array}\right),$$
$$T(x)=\left(\begin{array}{cc}1&2\\0&0\end{array}\right)=\left(\begin{array}{cc}1&0\\0&0\end{array}\right)+2\left(\begin{array}{cc}0&1\\0&0\end{array}\right),$$
$$T(x^2)=\left(\begin{array}{cc}0&2\\0&2\end{array}\right)=2\left(\begin{array}{cc}0&1\\0&0\end{array}\right)+
2\left(\begin{array}{cc}0&0\\0&1\end{array}\right).$$
So
$$[T]^{\gamma}_{\beta}=
\left(\begin{array}{ccc}0&1&0\\ 2&2&2\\ 0&0&0\\0&0&2\end{array}\right),$$
