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(1) The physical dimensions of vectors are followed throughout the book as in every book in mathematical physics. (2) The entries of a matrix are kernels, which come in two different forms: Scalar kernels and kernel functions. Kernels are equivalent to dyads. (3) Every system of linear algebraic equations is inherently entangled with its dual system: The former alone is not defined without the latter. (4) As a consequence of this entanglement, the given system of equations gives rise to a pseudo- and a dual pseudo-eigenequation. (5)With the aids of the r number of pseudo-and dual pseudo-eigenvectors where r is the rank of the matrix A of the system of equations (and its dual matrix), every matrix A (and its dual) can always be diagonalized (the generalized specrtal decomposition), leading to the fundamental decomposition of A and its natural inverse, yielding the least squares solution of the system of equations. (6) It is proved that hither-to-known Fourier series expansion of a function in a single space is in fact an infinite-dimensional system of algebraic equations from a domain space to a codomain space, and the formula for the Fourier coefficients is the solution of the system of equations. The Fourier matrix and its adjoint are composed of eigensolutions of a one-dimensional selfadjoint Sturm-Liouville system, which are biorthogonasl, not orthogonal as traditionally argued. The Fourier's formulas are propositions, for which we provide the proofs.