## The Range and Null Space of the Zero Transformation of Vector Spaces

## Problem 555

Let $U$ and $V$ be vector spaces over a scalar field $\F$.

Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$.

**(a)** Prove that $T:U\to V$ is a linear transformation.

(Hence, $T$ is called the **zero transformation**.)

**(b)** Determine the null space $\calN(T)$ and the range $\calR(T)$ of $T$.