Most people think of linear algebra as a tool for solving systems of linear equations. While it definitely helps with that, the theory of linear algebra goes much deeper, providing powerful insights into many other areas of math.
In this post I’ll explain a powerful and surprising application of linear algebra to another field of mathematics — calculus. I’ll explain how the fundamental calculus operations of differentiation and integration can be understood instead as a linear transformation. This is the “linear algebra” view of basic calculus.
Taking Derivatives as a Linear Transformation
In linear algebra, the concept of a vector space is very general. Anything can be a vector space as long as it follows two rules.