T/F: The Range of a Linear Transformation must be a subset of the domain.

Question

I am having difficulty understanding why it is false. As I understand, the range is the output of everything from the domain into the transformation function, or the result of the domain being outputted to the co-domain via the transformation. Essentially, I cannot think of a counterexample where the range of a linear transformation is not in the domain. I would appreciate if anyone could shed some light on this basic concept that I am struggling with.

Answer 1

Consider the following example of linear transformation,

$T:\mathbb R \to {\mathbb R}^2$

$T(x)= (x,0)$

Note that Range of $T$ is {$(x,0):x \in \mathbb R$} which is not a subset of the domain (i.e. $\mathbb R$).

Answer 2

Take $F\colon\mathbb{R}\longrightarrow\mathbb{R}[x]$ defined by $F(\lambda)=\lambda x$. Its range is a set of polynomials, and therefore it is not a subset of $\mathbb R$.