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Throughout this chapter, all vector spaces are assumed to be finite-dimensional unless otherwise noted.

In this chapter, we study vector spaces over arbitrary fields that have a bilinear form defined upon them.

In Chapter 9, we studied the basic properties of real and complex inner product spaces. Much of what we did does not depend on whether the space in question is finite or infinite-dimensional. However, as we discussed in Chapter 9, the presence of an inner product and hence a metric, on a vector space, raises a host of new issues related to convergence. In this chapter, we discuss briefly the concept of a metric space. This will enable us to study the convergence properties of real and complex inner product spaces.

Now that we have the necessary background on the topological properties of metric spaces, we can resume our study of inner product spaces without qualification as to dimension. As in Chapter 9, we restrict attention to real and complex inner product spaces. Hence will denote either ℝ or ℂ.

In the preceding chapters, we have seen several ways to construct new vector spaces from old ones. Two of the most important such constructions are the direct sum ⊕ and the vector space () of all linear transformations from to . In this chapter, we consider another very important construction, known as the .

Given a matrix consider the homogeneous system of linear equations = 0 It is of obvious interest to determine conditions that guarantee the existence of solutions to this system, in a manner made precise by the following definition.

In this chapter, we will study the geometry of a finite-dimensional vector space , along with its structure-preserving maps. .

Let be a finite-dimensional inner product space over , where = ℝ or = ℂ. Let us recall a definition.

In this chapter, we give a brief introduction to an area called the . This is a linear-algebraic theory used to study certain types of polynomial functions that play an important role in applied mathematics. We give only a brief introduction to the subject, emphasizing the algebraic aspects rather than the applications. For more on the umbral calculus, may we suggest , by Roman [1984]?