## Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$

## Problem 154

Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix}

x_1 \\

x_2

\end{bmatrix}\right )=\begin{bmatrix}

x_1-x_2 \\

x_1+x_2 \\

x_2

\end{bmatrix}$.

**(a) **Show that $T$ is a linear transformation.

**(b)** Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

**(c)** Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.