ABSTRACT
This chapter is about real and complex vector spaces equipped with an inner product or, more generally, a norm. An inner product can be usefully thought of as a generalization of the dot product defined on Rn whereas a norm assigns to each vector a “length.” 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 we can obtain a norm from an inner product, in particular, we prove that the Cauchy-Schwartz inequality holds for an inner product space as well as the triangle inequality. In section three we introduce several new concepts including that of an orthogonal sequence of vectors in an inner product space, an orthogonal basis, orthonormal sequence of vectors, and an orthonormal basis. We show how to obtain an orthogonal (orthonormal basis) of a 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 . We also define, for a basis BV in V , the basis, BV ′ , of V ′ dual to BV . For a linear transformation T from a finite-dimensional vector space V to a finitedimensional space W , we define the transpose transformation 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 transformation T ′ with respect to the bases BW ′ and BV ′ , which are dual to BW and BV , respectively. In section six, we make use of the transpose of a linear transformation T : V → W to define the adjoint transformation, T ∗ :W → V , of T . In section seven we
What You Need to Know
In order for the new material in this section to make sense you should have a fundamental understanding of the following concepts: a real vector space, a complex vector space, the space Rn, the space Cn, the space Mnn(R), and the space Mnn(C), the dot product on R.
