This question in my textbook asks me to determine if the following polynomial is in the null space and/or range space.
Let $h:P3 \to P4 $ be given by $p(x) \mapsto x\cdot p(x) $
Is $x^3$ in the null space or range space?
I said that it is not in the null space because $h(x^3) = x^4$ is not the zero polynomial of the codomain. But I do not understand how to determine if $x^3$ is in the range space? The book defines the range space of a homomorphism $ h: V \mapsto W $ as $\mathscr{R}(h) = \{h(\overrightarrow{v})|\overrightarrow{v}\in V\}$. My interpretation is that the range space is all vectors of the linear map such that v is part of the vector space. Is that correct?
If someone could please explain the concept of the range space, that would be great!
Thanks!