Is the sum of a linear and a non-linear transformation always non-linear? If not, then provide an example.
What about sum of two non-linear transformations?
In the latter case, I think there exist examples of non-linear transformations whose sum is linear.
Suppose $f:\mathbb{R}\to\mathbb{R}$ defined as $f(x)=x+1$ is non-linear (translation map) and $g:\mathbb{R}\to\mathbb{R}$ defined as $g(x)=-1$ is also non-linear (since,$g(0)=-1$) But $(f+g):\mathbb{R}\to\mathbb{R}$ is linear.
Plz clarify.
Yes: the sum of a linear transformation $f$ with a non-linear transformation $g$ is always non-linear. Let $h=f+g$. If $h$ was linear, then, since $g=h-f$, $g$ would be linear too. But we are assuming that it is not.
And your counterexample works, yes.