📂Functions

Identity Function

Definition¹

Given a set $X$, the following function $I_{X} : X \to X$ is called the identity function.

$$I_{X}(x) = x,\quad \forall x \in X$$

Explanation

The following notations are commonly used.

$$I,\quad \text{id},\quad \text{1}$$

Tangent vectors on a differentiable manifold are defined as follows in $\dfrac{d (f\circ \alpha)}{d t}$, where the function to be differentiated

$$f \circ \alpha = f \circ I \circ \alpha = f \circ \mathbf{x} \circ \mathbf{x}^{-1} \circ \alpha$$

can be decomposed like this, allowing the tangent vector to be represented with respect to any coordinate system $\mathbf{x}$ while making it independent of the choice of the coordinate system.

Example

Identity Matrix

If we think of the identity matrix as a linear transformation, it corresponds to the identity function. From an algebraic perspective, it is also the identity element with respect to matrix multiplication.*

$$I_{n\times n} = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix}$$