Given $c$ in $R$, define $T_c : R^n \to R$ by $T_c (x) = cx$ for all $x$ in $R^n$. Show that $T_c$ is a linear transformation and find its matrix.
I don't understand the question. For $T_c$ is $c$ the matrix and we are supposed to find $c$? Would the matrix just be $$ \begin{matrix} c & 0 & 0 ...0 \\ \end{matrix} $$
There is no "its matrix". There are infin ite matrices corresponding to different basis of $\;\Bbb R^n\;$ for $\;T\;$.
You have to apply $\;T\;$ on some basis of $\;\Bbb R^n\;$ and write down the result as a linear combination of the same basis (this is the easiest, most useful, case). Say, for example with the standard matrix $\;e\;$:
$$T_c(1,0,...,0):=(c,0,...,0)=c\cdot(1,0,...,0)+0\cdot(0,1,0,...,0)+\ldots+0\cdot(0,0,...,1)$$
and do the same for every element of the standard matrix. Now take the transpose of the coefficients matrix, and that is your matrix for this choice of basis :
$$[T_c]_e=\begin{pmatrix}c&0&0&\ldots&0\\ 0&c&0&\ldots&0\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&0&c\end{pmatrix}$$