Checking the linearity of a map

Question

Check whether or not the operator $Q$ which acts on functions in $V$ (some vector space) as $$Q:g(\lambda) \mapsto \lambda^5 g(\lambda)$$ is a linear map. Intuitively I know that it is not a linear map. However, I'm a bit confused with the notation on how one evaluates the expression to show a counterexample. How does one do that? What is this type of notation called?

Answer 1

You have two functions, $f(\lambda)$ and $g(\lambda)$. Each one is mapped individually to the function $\lambda^5 f(\lambda)$ and $\lambda^5 g(\lambda)$. For example, $f(\lambda)$ might be the function $\lambda\rightarrow\lambda^3+3$, so it gets mapped to the function $\lambda\rightarrow\lambda^8+\lambda^5$.

You should think about what the function $f(\lambda)+g(\lambda)$ gets mapped to. As a hint: This mapping IS linear, despite your intuition.