Exercise 6.2.1

For each of the following vector functions $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$, show that $T$ is a linear transformation and find the corresponding matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.

1. $T$ multiplies the $j^{\,\mathrm{th}}$ component of $\mathbf{v}$ by a non-zero number $b$.

2. $T$ adds $b$ times the $j^{\,\mathrm{th}}$ component of $\mathbf{v}$ to the $i^{\,\mathrm{th}}$ component.

3. $T$ switches the $i^{\,\mathrm{th}}$ and $j^{\,\mathrm{th}}$ components.