The Zero Transformation

# The Zero Transformation

 Definition: For any vector $\vec{x} \in \mathbb{R}^n$ where $0$ denotes the $m \times n$ zero matrix and $\vec{0} = (0, 0, ..., 0)$ denotes the $n$-component zero vector, the zero transformation $T_{0}: \mathbb{R}^n \to \mathbb{R}^m$ maps every vector (or point) $\vec{x}$ to $\vec{0}$, that is $T_0(x) = 0x = \vec{0}$.

For example, let's look at the zero transformation $T_0: \mathbb{R}^2 \to \mathbb{R}^2$. The image below illustrates the transformation on the vectors $\vec{a}, \vec{b}, \vec{c}$ (left) all of which are mapped to the zero vector in blue (right).

We can write the zero transformation $T_0: \mathbb{R}^2 \to \mathbb{R}^2$ as $T_0 (x_1, x_2) = (0, 0)$. Alternatively we can write this transformation in the matrix form of $w = Ax$ such that for any vector $\vec{x} = (x_1, x_2)$:

(1)
\begin{align} \quad \begin{bmatrix} 0\\ 0 \end{bmatrix} = \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x_1\\ x_2 \end{bmatrix} \end{align}

Thus the standard matrix for the zero transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$ is the $2 \times 2$ zero matrix as we defined.