The Identity Operator

# The Identity Operator

 Definition: For any vector $\vec{x} \in \mathbb{R}^n$ where $I$ represents the $n \times n$ identity matrix, the identity transformation $T_{I}: \mathbb{R}^n \to \mathbb{R}^n$ maps every vector $\vec{x}$ onto itself, that is $T_{I}(x) = Ix = \vec{x}$.

For example, consider the identity operator $T: \mathbb{R}^2 \to \mathbb{R}^2$. The vectors $\vec{a}, \vec{b}, \vec{c}$ (left) are mapped back onto themselves (right):

Additionally, we can summarize the image of any vector $\vec{x} = (x_1, x_2)$ with the following equations:

(1)
\begin{align} w_1 = x_1 + 0x_2 \\ w_2 = 0x_1 + x_2 \end{align}

In terms of matrix notation $w = Ax$, we get:

(2)
\begin{align} \quad \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_1\\ x_2 \end{bmatrix} \end{align}

Therefore we can clearly see that $I$ is the standard matrix for the identity operator.