Preimage

64. Preimage

For a transformation $𝑇 : ℝ^{𝑛} \to ℝ^{𝑚}$ and a subset $𝑆 \subseteq ℝ^{𝑚}$ , the preimage of $𝑆$ under $𝑇$ is the set of inputs that map into $𝑆$ :

𝑇^{- 1} (𝑆) = {\vec{𝑥} \in ℝ^{𝑛} | 𝑇 (\vec{𝑥}) \in 𝑆}

For a single point $\vec{𝑦} \in ℝ^{𝑚}$ , the preimage $𝑇^{- 1} (\vec{𝑦})$ may be empty, a single point, or an entire affine subspace — depending on $𝑇$ .

Example

Let $𝑇 : ℝ^{2} \to ℝ^{2}$ be defined by $𝐴 = [\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}]$ , so $𝑇 (\vec{𝑥}) = [\begin{matrix} 2 𝑥_{1} \\ 3 𝑥_{2} \end{matrix}]$ .

Preimage of $[\begin{matrix} 4 \\ 6 \end{matrix}]$ :

Solve $𝐴 \vec{𝑥} = [\begin{matrix} 4 \\ 6 \end{matrix}]$ :

\begin{matrix} 2 𝑥_{1} = 4 \Rightarrow 𝑥_{1} = 2 \\ 3 𝑥_{2} = 6 \Rightarrow 𝑥_{2} = 2 \end{matrix}

𝑇^{- 1} ([\begin{matrix} 4 \\ 6 \end{matrix}]) = {[\begin{matrix} 2 \\ 2 \end{matrix}]}

Preimage of $[\begin{matrix} 2 \\ 3 \end{matrix}]$ : similarly $𝑇^{- 1} ([\begin{matrix} 2 \\ 3 \end{matrix}]) = {[\begin{matrix} 1 \\ 1 \end{matrix}]}$ .

Preimage of the set $𝑆 = {[\begin{matrix} 4 \\ 6 \end{matrix}], [\begin{matrix} 2 \\ 3 \end{matrix}]}$ :

𝑇^{- 1} (𝑆) = {[\begin{matrix} 2 \\ 2 \end{matrix}], [\begin{matrix} 1 \\ 1 \end{matrix}]}

64.1. Properties

$𝑇^{- 1} (\emptyset) = \emptyset$
$𝑇^{- 1} (ℝ^{𝑚}) = ℝ^{𝑛}$
$𝑇^{- 1} (𝑆_{1} \cup 𝑆_{2}) = 𝑇^{- 1} (𝑆_{1}) \cup 𝑇^{- 1} (𝑆_{2})$
$𝑇^{- 1} (𝑆_{1} \cap 𝑆_{2}) = 𝑇^{- 1} (𝑆_{1}) \cap 𝑇^{- 1} (𝑆_{2})$
For a linear $𝑇$ : $𝑇^{- 1} (𝟎)$ is exactly the kernel of $𝑇$
For linear $𝑇$ and any $\vec{𝑦} \in im (𝑇)$ : $𝑇^{- 1} (\vec{𝑦}) = {\vec{𝑥}}_{𝑝} + ker (𝑇)$ for any particular solution ${\vec{𝑥}}_{𝑝}$ — an affine subspace parallel to the kernel

64.2. Connections

Image — the forward direction
Image of a subset — applying $𝑇$ to a set rather than $𝑇^{- 1}$
Kernel — preimage of the zero vector under a linear map