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 $T$.

(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*}
T(\mathbf{u}_1)&=\mathbf{0}_V, \quad T(\mathbf{v}_2)=\mathbf{0}_V, \\
T(\mathbf{u}_1+\mathbf{u}_2)&=\mathbf{0}_V, \quad T(r\mathbf{u}_1)=\mathbf{0}_V
\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
\[\calR(T)=\{\mathbf{0}_V\}.\]

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