ABSTRACT
In the first two sections of this chapter, we use orthogonality to complete our understanding of subspaces-especially the relationship between the span and perp of the same set. In the last two sections, we explore other topics that use orthogonality: First, we focus on orthonormal bases : bases comprised of mutually orthogonal unit vectors. Such bases are particularly useful, and they also have special geometric significance, for when they form the columns of a matrix, that matrix represents a distortion-free linear transformation, as we shall see. Lastly, we explain the Gram-Schmidt algorithm, which systematically replaces the vectors of any basis for a subspace S ⊂ Rn by those of an orthonormal basis.
