Linear algebra has a lot to say about linear transformations of vectors.
Which field studies non-linear transformations of vector spaces? What is a good introductory textbook on this matter?
Update: Differential geometry might be an answer: studying it.
Multivariable calculus is a mapping from multiple real numbers to a single real number. It is very different from many variables to many variables mapping that a linear algebra does. Complex analysis is a better example, because it captures the mapping of a pair-to-pair of the components of complex numbers. However, Cauchy–Riemann equations restrict derivatives over complex field, allowing a prove very strong theorems.
Is there something less constrained than the complex analysis with many-to-many variable mapping?
