Rank of linear operator on $M_{3\times 3}(R)$ The question asks the rank of linear operator $T$ on $M_{3 \times 3}(R)$ defined as: 

$T(P)=QP$

where Q is a square matrix of order 3 whose rank is 2.
I can't proceed with this as I have only come across operators on Euclidean n-dimensional space.

One definition of rank (of a linear map) is the dimension of the image.
Thus consider the linear map $T:M \rightarrow M$, where $M$ stands for $M_{3\times 3}(\Bbb R)$. Since we know $\dim M = 9$, to get the dimension of its image, it suffices to calculate the dimension of its kernel.
The map $T$ is defined by $T(P) = QP$, so $\ker(T) = \{P\in M: QP = 0\}$. If we write $P$ as $(P_1, P_2, P_3)$, with $P_i$ column vectors, then $P$ is in $\ker(T)$ if and only if $QP_i = 0$ for $i = 1, 2, 3$.
But $Q$ is a matrix of rank $2$, hence $QP_i = 0$ is equivalent to saying that $P_i$ is in the one dimensional kernel of $Q$ in $\Bbb R^3$.
Hence we conclude that $\dim \ker(T) = 3$, and thus the rank of $T$ is $9 - 3 = 6$.
