Linear Algebra: Lecture 34: complex inner product space, Hermitian conjugate and properties

Continuing Lecture 33, I fix the proof of coordinate independence of the projection to begin. Then we study complex inner product spaces briefly. Symmetric and self-adjoint linear transformations are discussed. I SHOULD have called self-adjoing linear transformations "Hermitian operators" hence the result that the e-values of a Hermitian operator are real. Finally I sketched the proof of the spectral theorem. I hope I've shown enough for you to understand the proof in Damiano and Little. Next time I'll emphasize a couple points I missed in this Lecture (why complex matters for the existence of an e-value, and the more general topic of allowed coordinate change). Then we begin work on generalized e-vector theory aka Chapter 6 of Damiano and Little.