In Section ?? we discuss orthonormal bases — bases in which each basis vector has unit length and any two basis vectors are perpendicular. We will see that the computation of coordinates in an orthonormal basis is particularly straightforward. We use orthonormality in Section ?? to study the geometric problem of least squares approximations (given a point and a subspace , find the point in closest to ) and in Section ?? to study the eigenvalues and eigenvectors of symmetric matrices (the eigenvalues are real and the eigenvectors can be chosen to be orthonormal). We present two applications of least squares approximations: the Gram-Schmidt orthonormalization process for constructing orthonormal bases (Section ??) and regression or least squares fitting of data (Section ??). The chapter ends with a discussion of the decomposition for finding orthonormal bases in Section ??. This decomposition leads to an algorithm that is numerically superior to Gram-Schmidt and is the one used in MATLAB.