\[
\displaylines{\text{Suppose a linear transformation }T:\mathbb{R}^{n}\to\mathbb{R}^{n}\text{ has the property that }T\left(\mathbf{u}\right)=T\left(\mathbf{v}\right)\\ \text{ for some pair of distinct vectors }\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}.\text{ Can }T\text{ map }\mathbb{R}^{n}\text{ onto }\mathbb{R}^{n}?\text{ Why or why not?}}
\]