We've about finished our journey through topics of linear algebra and applications that can be studied with them. As we've learned tools, we've seen a variety of ways to apply our learning.

The early chapters dealt extensively with applications of computer graphics and demonstrated the power and versatility of even simple mathematical operations. Such tools enable the computations to be done quickly, which can be useful and important in video games where quick calculations are necessary for user-defined movement of characters like the Viking in Figure 13.1. For example, Figure 13.1 (a) shows an underlying wire-frame model as created for movement while Figure 13.1 (b) removes its lines, enhancing the sense of realism and three dimensional look of the game.

The later chapters delved more into areas of data mining and analytics. In today's world the amount of information available and accessible about our world continually enlarges. Linear algebra is a means to harness this potential power. We've learned some basic methods and many others exist. However, it's important to know that methods continue to be developed, too.

And here is our last lesson. Even the field of linear algebra is far from linear. There are new twists and turns in how methods are used and new techniques that are created. The field cannot be accurately represented by a directed line segment. It continues to grow, encouraged, in part, by the many applications that derive from the usefulness of linear algebra. The boundaries of linear algebra also grow out of the creativity and ingenuity of mathematicians, like you. What applications interest you and came to mind as you read? Do new ideas present themselves as you complete this book? Those ideas can be your steps in the field and possibly take it into new directions. As we've seen, there are many applications and, as you may find as you create your ideas, there are many more to come.