Chapter 6: In this chapter, we explore the role of orthonormal (orthogonal and normalized) vectors in an inner-product space. Matrix representations of linear transformations with respect to orthonormal bases are of particular importance. They are associated with the notion of an adjoint transformation. We give a brief introduction to Fourier series that highlights the orthogonality properties of sine and cosine functions. In the final section of the chapter, we discuss orthogonal polynomials and the remarkable numerical integration rules associated with them.

A summary of essential definitions and useful formulas for vector, matrices and orthogonal basis sets employed in the book.

The eigenvalues and eigenfunctions of self-adjoint differential operators provide the basis functions with respect to which ordinary and partial differential equations can be solved.These methods are extensions of those used to solve linear systems of algebraic equations and ordinary differential equations.Eigenfunction expansions also provide the basis for advanced numerical methods, such as spectral methods, and data-reduction techniques, such as proper-orthogonal decomposition.

Vectors and matrices provide a mathematical framework for formulating and solving linear systems of algebraic equations, which have applications in all areas of engineering and the sciences.Solution methods include Gaussian elimination and the matrix inverse.Vectors play a key role in representing various quantities in mechanics as well as providing the bases for vector spaces.Linear transformations allow one to alter such vector spaces to ones that are more convenient for the task at hand.