Assumed knowledge

• Familiarity with functions, including injective, surjective and inverse functions;
• Familiarity with linear equations, including solving simultaneous equations.
• Familiarity with vectors and $2 \times 2$ matrices, including position vectors and matrix multiplication and inverses;
• Familiarity with plane Euclidean geometry.

Motivation

As in many aspects of life, it's often useful in mathematics to see things from different perspectives. Changing your point of view involves some kind of transformation.

You see geometric transformations every minute of every day: whenever you walk around an object, you see a rotation. When you see an image of an object, it is a projection. When you move an object, you see a translation. When you look in a mirror, you see a reflection.

In this module we focus on transformations of the plane. Plane transformations encompass a great deal of geometry, including rotations, reflections, projections, translations, and many other operations.

We mostly focus on transformations of a very particular type: linear transformations. Linear transformations form a narrow, but useful, class of plane transformations. We will see how such transformations can be described by some relatively simple algebra.

Studying plane transformations brings together algebra and geometry in a coherent and elegant way. We will see that the algebra of linear functions and matrices efficiently describes a wide array of geometric transformations.

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