The matrix of a linear transformation

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.

Assume that $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is a linear transformation, and that $\mathbf{u}_{1},\ldots,\mathbf{u}_{n}$ is a basis of $\mathbb{R}^{n}$. For all $i=1,\ldots,n$, let $\mathbf{v}_{i}=T(\mathbf{u}_{i})$. Let $A$ be the matrix that has $\mathbf{u}_{1},\ldots,\mathbf{u}_{n}$ as its columns, and let $B$ be the matrix that has $\mathbf{v}_{1},\ldots,\mathbf{v}_{n}$ as its columns. Show that $A$ is invertible and the matrix of $T$ is $BA^{-1}$.

Suppose $T$ is a linear transformation such that

Find the matrix of $T$. Hint: use Exercise 7.

Suppose $T$ is a linear transformation such that

Find the matrix of $T$. Hint: use Exercise 7.

Find the matrix for $T(\mathbf{w})=\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$, where $\mathbf{v}=\left[1,-2,3\right]^{T}$.

Find the matrix for $T(\mathbf{w})=\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$, where $\mathbf{v}=\left[1,5,3\right]^{T}$.