derivative as a linear operator

Intuitively, computing a derivative is a linearization process. ¹

Building on this intuition, it is possible to show that derivative satisfies the defining properties of a linear transformation: additivity and homogeneity. Specifically, for any functions and , the differentiation operator obeys the following relation in D-notation:

This insight helps determine the dimensions of a derivative. A linear operator from a n-dimensional Euclidean vector to a m-dimensional vector in can be represented as an matrix. And as a result,

• the derivative of a function with n input and 1 output is an -dimensional gradient vector
• the derivative of a multi-variable function with input and output, if exists, is an Jacobian Matrix ¹

Footnotes

1. So You Think You Know How to Take Derivatives? | Steven Johnson | ASE60 - YouTube ↩ ↩²