Here, roughly, are the topics we've covered since the second midterm:

• Inner products
  • norms
  • geometric inequalities
  • orthonormal bases
  • Gram matrices, orthonormalization
  • perpendicular space
  • orthogonal projection
• Operators on inner product spaces
  • adjoint -- formal definition, computation
  • self-adjoint operators
  • normal operators
  • spectral theorem (real, complex, self-adjoint, normal)
  • isometries
• Canonical forms
  • minimal polynomial, characteristic polynomial
  • generalized eigenvectors
  • nullspaces, images of iterates
  • block decompositions
  • Jordan normal form