Introduction to Matrices - 3D Math Primer for Graphics and Game Development - page 130

take simple transformations and derive more complicated transformations
through matrix concatenation; more on this in Section 5.6.
Before we move on, let's review the key concepts of Section 4.2.

• The rows of a square matrix can be interpreted as the basis vectors

of a coordinate space.

• To transform a vector from the original coordinate space to the new

coordinate space, we multiply the vector by the matrix.

• The transformation from the original coordinate space to the coordi-

nate space defined by these basis vectors is a linear transformation.
A linear transformation preserves straight lines, and parallel lines re-
main parallel. However, angles, lengths, areas, and volumes may be
altered after transformation.

• Multiplying the zero vector by any square matrix results in the zero

vector. Therefore, the linear transformation represented by a square
matrix has the same origin as the original coordinate space—the
transformation does not contain translation.

• We can visualize a matrix by visualizing the basis vectors of the co-

ordinate space after transformation. These basis vectors form an 'L'
in 2D, and a tripod in 3D. Using a box or auxiliary object also helps
in visualization.

4.3 The Bigger Picture of Linear Algebra

At the start of Chapter 2, we warned you that in this topic we are focus-
ing on just one small corner of the field of linear algebra—the geometric
applications of vectors and matrices. Now that we've introduced the nuts
and bolts, we'd like to say something about the bigger picture and how our
part relates to it.
Linear algebra was invented to manipulate and solve systems of linear
equations. For example, a typical introductory problem in a traditional