ABSTRACT
This chapter is about inner products on real and complex vector spaces. An inner product can be usefully thought of as a generalization of the dot product defined on Rn. In the first section, we define the concept of an inner product, give several examples, and investigate basic properties. In section two, we indicate how an inner product allows us to introduce notions of length, orthogonality and other aspects of geometry. We will prove the Cauchy-Schwartz inequality, which then allows us to introduce angles between vectors and to prove the triangle inequality. In section three, we introduce several new concepts, including an orthogonal sequence of vectors in an inner product space, an orthogonal basis, and orthonormal sequence of vectors and an orthonormal basis. We show how to obtain an orthogonal (orthonormal basis) of an finite dimensional inner product space when given a basis of that space. In section four, we prove that if U is a subspace of an finite dimensional inner product space (V, 〈 , 〉) then V is the direct sum of U and its orthogonal complement. This is used to define the orthogonal projection onto U. In section five, we define the dual space V ′ of a finite dimensional vector space V and for a basis B in V the notion of the basis of V ′ dual to BV . For a linear transformation T from a finite dimensional vector space V to a finite dimensional space W, we define the transpose operator T ′ from W ′ to V ′. We investigate the relationship between that matrix of T with respect to bases BV and BW and the matrix of the transpose operator T ′ with respect to the bases dual to BW and BV . Finally, in the concluding section, we make use of the transpose to define the adjoint of a linear transformation.
