91Ó°ÊÓ

Determine whether the linear transformation \(T\) is \((a)\) one-to-one and \((b)\) onto. $$T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \text { defined by } T\left[\begin{array}{l}x \\\y\end{array}\right]=\left[\begin{array}{l}2 x-y \\\x+2 y\end{array}\right]$$

Let \(T: V \rightarrow W\) be a linear transformation between two finite- dimensional vector spaces. (a) Prove that if \(\operatorname{dim} V<\operatorname{dim} W\), then \(T\) cannot be onto. (b) Prove that if \(\operatorname{dim} V>\operatorname{dim} W\), then \(T\) cannot be one-to-one.