Answer
See explanation
Work Step by Step
(b). If $T$ is one-to-one, what can you say about $m$ and $n ?$
Concepts
Definition of linear transformation
Definition of one-to-one mapping Definition of onto mapping Theorems 4 and 12
Solve (a)
If $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ maps $\mathbb{R}^{n}$ onto $\mathbb{R}^{m},$ then its standard matrix $A$ has a pivot column in each row.
So $A$ must have at least as many columns as rows. In other words, $m \leq n$
Solve
When $T$ is one-to-one, $A$ must have a pivot in each column so $m \geq n$