Chapter 1 - Linear Equations in Linear Algebra - 1.9 Exercises - Page 80: 35

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(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 $m \geq n$