Finite Dimensional Vector Spaces
Vector spaces, subspaces, direct sums, bases, dimension.

Inner-Product Spaces
Inner-products, norms, orthonormal bases, Gram-Schmidt, orthogonal projection and minimization, linear functionals and adjoints.

Linear Maps
Null spaces and ranges, matrices, eigenvalues and eigenspaces, similarity, self-adjoint and normal operators, spectral theorem, isometries, polar and singular value decompositions, characteristic and minimal polynomial, Cayley's Theorem, Jordan form, trace and determinant of operators.

References:

Linear Algebra Done Right, 2^nd edition, Sheldon Axler, Springer.
Linear Algebra, 4^th edition, Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Prentice Hall.