Inverse functions and transformations

You can use a transformation/function to map from one set to another, but can you invert it? In other words, is there a function/transformation that given the output of the original mapping, can output the original input (this is much clearer with diagrams). This tutorial addresses this question in a linear algebra context. Since matrices can represent linear transformations, we're going to spend

Introduction to the inverse of a function

Proof: Invertibility implies a unique solution to f(x)=y

Surjective (onto) and injective (one-to-one) functions

Relating invertibility to being onto and one-to-one

Determining whether a transformation is onto

Exploring the solution set of Ax = b

Matrix condition for one-to-one transformation

Simplifying conditions for invertibility

Showing that inverses are linear