Standard Matrix vs. Identity Matrix

Question

Is there a difference between identity and standard matrices? Or are they are just different notations to distinguish between regular matrices and linear transformations?

Answer 1

The $n\times n$ identity matrix is always $I=[\delta_{ij}]$ where $\delta_{ij}=\begin{cases} 1 & \text{if } i=j \\ 0 & \text{otherwise}\end{cases}$. For example the 3x3 identity matrix is $I=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

The standard matrix of a linear transformation $T:\mathbb{R}^n \to \mathbb{R}^m$ is an $m \times n$ matrix $A$ with the property that for all vectors $x$ evaluating the linear transformation is the same as multiplying by the matrix. That is $T(x)=Ax$.

So it is possible for an identity matrix to be a standard matrix, but it's not always the case.