Show That a 2D Linear Transform $ T \left( \cdot \right) $ Is Homogeneous
The idea here is that any Linear Operator is Homogeneous though not every Homogeneous Operator is Linear.

The classic proof indeed is to build the zero term as a sum (Difference) of 2 other elements in the domain. Then use Linearity to show it must be zero:

$$ T \left( 0, 0 \right) = T \left( {m}_{1} - {m}_{1}, {n}_{1} - {n}_{1} \right) = T \left( {m}_{1}, {n}_{1} \right) - T \left( {m}_{1}, {n}_{1} \right) = 0 $$ 


  Maybe the better question to ask would be to prove if a transformation is heterogeneous, is it linear?


As I wrote above, if an operator is heterogeneous is must not be linear.

Think of the simplest Linear Operator in 1D: $ T \left( x \right) = a x $. Clearly it is a linear function and indeed homogeneous.
Let's look on Affine Function: $ T \left( x \right) = a x + b $. Usually this is also called Linear Function while it is not unless $ b = 0 $. Why? It is not heterogeneous.

It is easy to see that $ T \left( {x}_{1} - {x}_{1} \right) \neq T \left( {x}_{1} \right) - T \left( {x}_{1} \right) $.