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Chapter3Linear Transformations and Matrix Algebra

Primary Goal

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.

A xy B = f ( θ , φ , ψ ) θ φ ψ

Define a transformation f as follows: f ( θ , φ , ψ ) is the ( x , y ) position of the hand when the joints are rotated by angles θ , φ , ψ , respectively. The output of f 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 3.1, we will consider the equation b = Ax as a function with independent variable x and dependent variable b , 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 3.2). In Section 3.3 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 3.4). 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 3.5.