Linear Algebra-Isomorphic linear transformation

In order to determine whether or not a transformation is both linear and isomorphic we must check to see if it meets the following criteria:

In order to determine linearity, usually the following two properties are sufficient:
T(f+g)=T(f)+T(g)
T(kf)=kT(f)

In order to determine Isomorphism, we must check to see if the kernel of the transform is 0. Which also means for a given transformation T from V to W, the image is W, fulfilling the rest of the requirements for isomorphism. Thus we can just check to see if the kernel is 0 or non-zero.

Example:
T(M)=M |1 2|
             |3 6|
M=|a b|
     |c d|
|a+3b 2a+6b|
|c+3d 2c+6d|
=
|0 0|
|0 0|

We can then separate this into solving for a,b which will be replicated for c,d
|1 3 0|
|2 6 0|
a = -3b

This makes M =
|-3b b|
|-3d d|
Thus the kernel is nonzero, and so this is not an isomorphic transformation.