Let $V$ be the vector space of all real $2 \times2$ matrices. Let $T: V \to \mathbb R^2$ be the map defined by $T(\left(\begin{array}{crc} a_{11} & a_{12}\\ a_{21} & a_{22} \\ \end{array}\right))=\left(\begin{array}{crc} 2a_{11}-a_{21} & 2a_{12}-a_{22}\\ \end{array}\right)$
Find bases for the kernel and image of $T$.
If the linear transformation was something like $T(x,y)=(2x-y,x+z)$ I'd be able to find bases for the kernel and image, but the format of the question seems different to me and I'm feeling very puzzled. Would someone mind showing me how to do this problem?
Edit: Also how would you find the matrix of $T$ with respect to the standard basis of $V$ and the standard basis of $\mathbb R^2$?