Domain

59. Domain

For a function $𝑓 : 𝑋 \to 𝑌$ , the domain is the set $𝑋$ of inputs.

𝑋 = domain (𝑓) = set of all valid inputs

For a linear transformation $𝑇 : ℝ^{𝑛} \to ℝ^{𝑚}$ , the domain is $ℝ^{𝑛}$ .

59.1. Domain vs codomain vs image

Domain — where the inputs live ( $𝑋$ )
Codomain — where outputs are allowed to live ( $𝑌$ )
Image / Range — where outputs actually land (a subset of the codomain)

Example

$𝑓 (𝑥) = 𝑥^{2}$ from $ℝ$ to $ℝ$ :

Domain: $ℝ$
Codomain: $ℝ$
Image: $[0, \infty)$ — only non-negative outputs

The image is a strict subset of the codomain (negatives like $- 3$ are in the codomain but not the image).

59.2. For matrices

If $𝑇 (\vec{𝑥}) = 𝐴 \vec{𝑥}$ with $𝐴$ an $𝑚 \times 𝑛$ matrix, then:

Domain = $ℝ^{𝑛}$ (number of columns of $𝐴$ )
Codomain = $ℝ^{𝑚}$ (number of rows of $𝐴$ )
Image = column space of $𝐴$ ⊆ $ℝ^{𝑚}$

59.3. See also

Codomain
Image — what lands in the codomain
Linear Transformation