# 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 $T$.

## Proof.

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

Let $\mathbf{u}_1, \mathbf{u}_2\in U$ and $r$ be a scalar, that is, $r\in \F$.
It follows from the definition of $T$ that
\begin{align*}
\end{align*}
since $\mathbf{u}_1+\mathbf{u}_2, r\mathbf{u}_1\in U$.
Hence we have
\begin{align*}
T(\mathbf{u}_1+\mathbf{u}_2)&=\mathbf{0}_V=\mathbf{0}_V+\mathbf{0}_V=T(\mathbf{u}_1)+T(\mathbf{u}_2)\\
T(r\mathbf{u}_1)&=\mathbf{0}_V=r\mathbf{0}_V=rT(\mathbf{u}_1).
\end{align*}
Since these equalities holds for all $\mathbf{u}_1, \mathbf{u}_2\in U$, and $r\in \F$, the map $T:U\to V$ is a linear transformation.

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

The null space $\calN(T)$ of $T$ is, by definition,
\begin{align*}
\calN(T)=\{\mathbf{u}\in U \mid T(\mathbf{u})=\mathbf{0}_V\}.
\end{align*}
Since $T(\mathbf{u})=\mathbf{0}_V$ for every $\mathbf{u}\in U$, we obtain
$\calN(T)=U.$

The range $\calR(T)$ of $T$ is, by definition,
$\calR(T)=\{\mathbf{v} \in V \mid \text{there exists } \mathbf{u}\in U \text{ such that } T(\mathbf{u})=\mathbf{v}\}.$

Since every vector of $U$ is mapped into $\mathbf{0}_V$, we have
$\calR(T)=\{\mathbf{0}_V\}.$

• A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$ Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$. Consider a linear transformation $T:U\to V$. Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).   Hints. You may use the folowing facts. A linear […]
• Linear Properties of Matrix Multiplication and the Null Space of a Matrix Let $A$ be an $m \times n$ matrix. Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$. Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$. Then find $A\mathbf{w}$.   Hint. Recall that the null space of an […]
• Dimension of Null Spaces of Similar Matrices are the Same Suppose that $n\times n$ matrices $A$ and $B$ are similar. Then show that the nullity of $A$ is equal to the nullity of $B$. In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of […]