I'd like an intuitive answer to the following question, and not the usual rule of checking:
$$T(u+v)=T(u)+T(v) \quad\text{and}\quad T(cu)=cT(u)$$
When there is a matrix transformation such as
$$ \begin{pmatrix} a & x & d \\ b & y & e \\ c & z & f \\ \end{pmatrix} $$ for any relevant vector X, then it is clear that the transformation is linear. But when is it the case that such a matrix exists but the transformation is not linear? And why? Does linear dependence has any relationship with this? I am clearly asking if there's a matrix transformation which is non linear.
If you define you transformation with something like $T(u)= Au$, with $A$ a matrix, then the transformation is always linear, no matter what $A$ you pick. The problem arise when you define it with different mathematical objects, such as an arbitrary formula: $$ T(x,y) = x^2- x + 4y. $$ Then it is quite easy to check if the given formula makes a linear o a non-linear transformation, for example by checking additivity: $$ T(x+x', y+y') = (x+x')^2 - x-x' +4y + 4y'\\ = x^2 + x'^2 + 2xx' - x - x' + 4y + 4y' \neq T(x,y) + T(x',y') = x^2 -x+4y + x'^2 -x' +4y'. $$ A good heuristic is: a linear function is usually a composition of 'primitive' linear functions (like sum and product). If squares, exponentials or other stuff appears, it probably isn't linear.