### The Jacobian of a linear map

This is a small blog post. What is the Jacobian of a linear map? Say I have an $n\times n$ matrix- call it $A$. Also, I have a linear map $L:\Bbb{R}^n\to\Bbb{R}^n$ which is given by $v\to Av$. What is the Jacobian of $L$? This is a question that has confused me before.

The function $f_i$ takes a vector $(x_1,x_2,\dots,x_n)$, and maps it to $a_{i1}x_1+a_{i2}x_2+\dots+a_{in}x_n$. The derivatives of this function with respect to the $x_k$‘s are just $a_{ik}$. Hence, the Jacobian of a linear map $L$ is the $L$ itself.