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

• Linear transformations
  • abstract definitions
  • image, nullspace, dimension theorems
  • matrices, bases, change of basis
  • matrix multiplication and composition
  • linear transformations and systems of equations; row space, column space, rank, nullity, etc.
  • invertible linear transformations, isomorphisms
  • linear operators
• Invariant subspaces
  • invariant spaces
  • eigenvalues and eigenvectors
  • independence of eigenvectors
  • polynomials and operators
  • upper-triangular matrices and suitable bases
  • diagonal matrices and bases of eigenvectors
  • eigenspace is null space of suitable transformation
  • characteristic polynomials
• Inner products
  • basic definitions