Answer
$T$ does not map $\mathrm{R}^{n}$ onto $\mathrm{R}^{n}$.
Work Step by Step
Let $\mathrm{A}$ be the standard matrix of $\mathrm{T}$.
$\mathrm{T}$ is not one-to-one, so by Th.12b of chapter 1,
the columns of A are linearly dependent.
A is a n$\times$n matrix, so the columns of $A$ do not span $\mathrm{R}^{n}$.
Applying Th.12a of chapter 1,
$T$ does not map $\mathrm{R}^{n}$ onto $\mathrm{R}^{n}$.
