Page 339 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

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CHAPTER

General Vector Spaces

I N T R O D U C T I O N : In the last chapter we generalized vectors from 2- and 3-space to vectors in n-space. In this chapter

we shall generalize the concept of vector still further. We shall state a set of axioms that, if satisfied by a class of objects, will
entitle those objects to be called “vectors.” These generalized vectors will include, among other things, various kinds of
matrices and functions. Our work in this chapter is not an idle exercise in theoretical mathematics; it will provide a powerful
tool for extending our geometric visualization to a wide variety of important mathematical problems where geometric intuition
would not otherwise be available. We can visualize vectors in and as arrows, which enables us to draw or form mental
pictures to help solve problems. Because the axioms we give to define our new kinds of vectors will be based on properties of
vectors in and , the new vectors will have many familiar properties. Consequently, when we want to solve a problem
involving our new kinds of vectors, say matrices or functions, we may be able to get a foothold on the problem by visualizing
what the corresponding problem would be like in and .

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