Explore the applications of 2x2 matrices in this comprehensive 55-minute lecture from the Wild Linear Algebra series. Delve into the interpretation of 2x2 matrices as linear transformations of the plane, analyze rotations including rational formulations, and examine the combination of rotations and reflections. Gain insights into the connections between linear algebra and calculus, introducing the concept of derivatives as linear transformations. Learn through detailed examples, proofs, and exercises covering topics such as matrix-vector multiplication, linearity properties, basis vector transformations, rotation matrices, trigonometric identities, rational parametrization, reflections, and composition of linear transformations. Conclude with an introduction to linear approximations of non-linear maps and the derivative matrix, bridging the gap between linear algebra and differential calculus.