Friday, March 29, 2013

Non-linear Additive Maps

While teaching an introductory linear algebra course, a colleague noticed that most of the examples of additive maps he gave turned out to be linear.  He asked whether I could think of a map which was additive, but not linear.  In a general context, the question was to find a ring $R$, $R$-modules $ V$ and $ W$, and a map $ f \colon V \to W$ such that $ f $ is a group homomorphism, but not an $ R$-module morphism, i.e, $ f(x+y)=f(x)+f(y)$ for all $ x,y \in V,$ but there is some $ r \in R$ and $ z \in V$ such that $ f(r \cdot z) \neq r \cdot f(z)$.

One example that came to mind was viewing $ \mathbb{C}$ as a vector space over itself and taking $ f \colon \mathbb C \to \mathbb C$ to be the reflection $ f(a+bi):=b+ai$.  This map is readily seen to be a group endomorphism of $ (\mathbb C, +)$, but it does not commute with rotation counter-clockwise by by $ \pi/2$ radians, which is just multiplication by $ i$.  In particular, $ f(i \cdot 1)=f(i)=1 \neq -1 = i^2 = i \cdot f(1)$.