# 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
$\calR(T)=\{\mathbf{0}_V\}.$

### More from my site

• 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 […]
• An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$. A linear transformation $T:\R^n \to \R^n$ is called orthogonal transformation if for all $\mathbf{x}, \mathbf{y}\in […] • Linear Properties of Matrix Multiplication and the Null Space of a Matrix Let$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)$of […] • Differentiation is a Linear Transformation Let$P_3$be the vector space of polynomials of degree$3$or less with real coefficients. (a) Prove that the differentiation is a linear transformation. That is, prove that the map$T:P_3 \to P_3$defined by $T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)$ for any$f(x)\in […]
• Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation Let $V$ be the vector space of all $n\times n$ real matrices. Let us fix a matrix $A\in V$. Define a map $T: V\to V$ by $T(X)=AX-XA$ for each $X\in V$. (a) Prove that $T:V\to V$ is a linear transformation. (b) Let $B$ be a basis of $V$. Let $P$ be the matrix […]
• Find a Value of a Linear Transformation From $\R^2$ to $\R^3$ Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are unit vectors of $\R^2$ and […]
• Linear Transformation to 1-Dimensional Vector Space and Its Kernel Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation. Prove the followings. (a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$. (b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the […]

#### You may also like...

##### Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism

Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula \[T\left(\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}...

Close