Eigenfunctions and Continuous Bounded 1D Wave Equation

This lecture covers the solution to the discretized system by identifying a basis of eigenvectors and solving the wave equation. It then explores the continuous wave problem, finding eigenfunctions of the differential operator and discussing the similarities and differences between continuous and discrete systems. The lecture delves into the orthogonality of eigenfunctions, the symmetric properties of the operator, and the harmonic oscillator equation. It concludes with the decomposition of the solution on the basis of eigenfunctions and the implications for solving linear differential equations. The lecture emphasizes the importance of linear algebra concepts in understanding the eigenmodes of vibration and introduces the notion of a Hilbert space for further analysis.