>>12442131First third: vector space definition, linear independence, span, dimension, subspace, important differences from "direction and magnitude" notion in R^2 and R^3, plenty of examples of particular vector spaces, linear maps, completely specifying a linear map by its action on the basis, matrix notation and comparison to representing f(x)=ax as [a]. range and null space, injectivity and surjectivity. invariant subspaces and eigenvalues.
Second third: determinants and their properties motivated as a linear dependence test in the target space. invertability. normalized-column determinants as an "overlap measurement." geometric meaning of the determinant in R^2 and R^3. reintroducing "direction and magnitude" for "R^n-like" spaces and what it changes vs the general vector space. inner product, induced norms, induced metric and distance. geometric consequences of the IP (parallelogram equality, polarization identity, pythagorean equality.) inner product as a similar "overlap measurement." orthonormal bases. self-adjoint, unitary, and normal operators. spectral theorem. change of basis. "eigenbasis" from the invariant subspaces as the most natural basis and SVD.
Final third: choose further topics such as jordan canonical form, general bilinear forms, definition of a group plus linear representation theory and its applications to spectroscopy/molecular vibrations, tensors with some way to work in relativity, or something else representing a big concrete payoff for all the theory, ideally aligned with student interest
