The Jacobian of a linear map

by ayushkhaitan3437

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.