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Numerical Linear Algebra with Applications by William Ford

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Chapter 6

Orthogonal Vectors and Matrices

Abstract

This chapter reviews two- and three-dimensional vectors, including computing the length of a vector. The inner product of two vectors is defined and its major properties are developed. These include the well-known fact that the inner product of two- or three-dimensional vectors u and v equals length(u) × length(v) × cos(angle between u and v). This result can be used to determine if two vectors are orthogonal (perpendicular) or parallel. The orthogonal matrix is defined as a matrix for which its inverse is its transpose and its relation to the inner product is developed; in particular, a matrix is orthogonal if and only if its columns are orthogonal and of unit length (orthonormal). The chapter ...

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