Find out if the transformation is linear, if so, determine whether it is an isomorphism. 
Hi, I know how to show that this is a linear transformation. But I am not sure how to figure out if it isomorphic. I tried performing the transformation with M = (a, b, c, d).
When I multiply this out I get a matrix (-2a, -3b, -c, -2d). 
Is this isomorphic because the basis of this space is 4 dimensional: (-2, 0 , 0 , 0) ; (0, -3, 0 , 0) ; (0, 0 , -1, 0) ; ( 0, 0 , 0 , -2)? Meaning, if the matrix I got when multiplying this out was something like (a, 0 , b, c), it would signify a kernel of 1, and thus would not be isomorphic?
Thanks! 

Hint: consider  $M $ a generic matrix $2×2$ and show that $\ker T=0$ then $M=0$
Edit:
Consider $$M=
        \begin{matrix}
        a & b  \\
        c & d  \\
        \end{matrix}
$$ 
To find $\ker T$ $T(M)= null matrix$
If find that $a=b=c=d=0$ $M$ is the null matrix and the $\ker $ is zero
