Topics In Linear Algebra and Its Applications - STEM2

where n and m have no common factor. Then32 = n2m 2,3m 2 = 2n 2 ,so n 2 and thus n must have a factor of 3. Hence m 2 and hence m must havea factor of 3. This contradicts our assumption.See Angus Taylor Introduction to Functional Analysis for a treatmentof Hamel basis.33 Numerical Linear AlgebraSee Golub and Van Loan, and books on Matlab.34 Quantum MechanicsQuantum Mechanics was treated by John Von Neumann in terms of HilbertSpaces and Linear Operators in Hilbert Space, and involving eigenvalues andeigenvectors of such spaces of functions. One runs into things like delta functions,which are approximations to functions, but not actually functions atall. Thus we use mathematical objects called distributions. These distributionsare functionals and not to be confused with probability distributions.Von Neumann made great contributions to the study of Banach Algebras.35 The Schrodinger Wave EquationFrom the photoelectric effect, the energy of a photon isE = hν,where h is Planck’s constant, and ν is the frequency of the photon. Fromrelativity theory, the energy of a particle is given by the equationE 2 = p 2 v 2 + m 2 c 4 ,43