Here is something I never thought about.
Additivity and homogeneity are conditions for a linear transformation for a reason. That is, it's possible for a function to be homogeneous but not additive and to be additive but not homogeneous. For example, consider \(\varphi\colon\mathbb{R}^2\to\mathbb{R}\) defined by \(\varphi(x,y)=\sqrt{xy}\). This is homogeneous but not additive. Alternatively, for \(\varphi\colon\mathbb{C}\to\mathbb{C}\), we can consider the map \(a+bi\mapsto b+ai\). This is additive but not homogeneous! In fact, it can be shown that an additive but not homogeneous function exists \(\mathbb{R}\to\mathbb{R}\), though the proof is nonconstructive and uses the axiom of choice. Meanwhile, I learned the proof of the Fundamental Theorem of Algebra just a few weeks ago. Very tricky, but very interesting. If anything I think I can get a cool diagram out of it.
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