Chapter 8
Matrices and LinearMatrices and Linear
Transformations
In Chapter 7, we investigated some of the basic mathematical properties of matrices. We also
developed a geometric understanding of matrices and their relationship to coordinate space trans
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formations in general. This chapter discusses this relationship between matrices and linear
transformations in more detail.
To be more specific, this chapter is concerned with expressing linear transformations in 3D
using 3×3 matrices. Linear transformations were introduced in Section 7.2. Recall that one impor
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tant property of linear transformations is that they do not contain translation. A transformation
that contains translation is known as an affine transformation. Affine transformations in 3D cannot
be implemented using 3×3 matrices. We will see a formal definition of affine transformations in
Section 8.8.2, and we will learn how to use 4×4 matrices to represent affine transformations in
Section 9.4.3.
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This chapter discusses the implementation of linear transformations using matrices. It is
divided into eight sections.
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Section 8.1 describes the relationship between transforming an object and transform-
ing the coordinate space used to describe the object.
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Sections 8.2 through 8.6 describe the primitive linear transformations of rotation,
scaling, orthographic projection, reflection, and shearing, respectively. For each
transformation, examples and equations are given in 2D and 3D.
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Section 8.7 shows how a sequence of primitive transformations may be combined
using matrix multiplication to form a more complicated transformation.
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Section 8.8 discusses various interesting categories of transformations, including
linear, affine, invertible, angle-preserving, orthogonal, and rigid body transforms.