##### Primary Goal

Learn about linear transformations and their relationship to matrices.

Learn about linear transformations and their relationship to matrices.

In practice, one is often lead to ask questions about the geometry of a *transformation*: a function that takes an input and produces an output. This kind of question can be answered by linear algebra if the transformation can be expressed by a matrix.

Suppose you are building a robot arm with three joints that can move its hand around a plane, as in the following picture.

Define a transformation as follows: is the position of the hand when the joints are rotated by angles respectively. The output of tells you where the hand will be on the plane when the joints are set at the given input angles.

Unfortunately, this kind of function does not come from a matrix, so one cannot use linear algebra to answer questions about this function. In fact, these functions are rather complicated; their study is the subject of inverse kinematics.

In this chapter, we will be concerned with the relationship between matrices and transformations. In Section 4.1, we will consider the equation as a function with independent variable and dependent variable and we draw pictures accordingly. We spend some time studying transformations in the abstract, and asking questions about a transformation, like whether it is one-to-one and/or onto (Section 4.2). In Section 4.3 we will answer the question: “when exactly can a transformation be expressed by a matrix?” We then present matrix multiplication as a special case of composition of transformations (Section 4.4). This leads to the study of *matrix algebra*: that is, to what extent one can do arithmetic with matrices in the place of numbers. With this in place, we learn to solve matrix equations by *dividing* by a matrix in Section 4.5.