Linear transformations of finite dimensional vector spaces are studied in a semi-abstract setting. The emphasis is on topics and techniques which can be applied to other areas, e.g. bases and dimension, matrix representation, linear functionals, duality, canonical forms, vector space decomposition, inner products and spectral theory.

Alexander Borisov

MWF 10:00-10:50 am in 704 Thackeray Hall.

Th 10:00-10:50 am in 704 Thackeray Hall. Recitations Instructor: Jonathan Holland.

1) To help the students prepare for the Linear Algebra Preliminary Examination.

2) To help the students formulate their goals to get the most from their graduate school training.

There will be a midterm and a final examination, and the overall grade in the course will be primarily determined by the performance on these two tests. Additionally, there will be occasional quizzes and graded homework that will have some effect on the overall grade. It is a student's responsibility to attend every lecture, take notes and go over the notes before the next lecture to fully understand the material covered. If for any reason you can not attend a lecture, please get the notes from another student in the class. You should also notify the instructor of your absense. If needed, ask the instructor for help with the missed material. Some homework problems will be assigned regularly, but not collected. It is your responsibility to do them and check with other students whether or not you got them right. If in doubt, seek help from the instructor or the recitation instructor. Recitation may have separate homework.

Note: the topics and the order of coverage is subject to change. Many of the topics will take several classes to cover.

1) Linear vector spaces, subspaces. Linear maps.

2) Equivalence relations. Quotient spaces. Kernel and Image of a linear map. Injective and surjective maps. Short exact sequences.

3) Linear independence of vectors. Span of a collection of vectors. Basis. Dimension.

4) Review of matrices. Algebraic operations on matrices. Invertible matrices. Notion of the determinant.

5) Matrix of a linear map. Composition of linear maps. Change of basis.

6) Determinant of a matrix and its geometric meaning. Trace of a matrix.

7) Eigenvectors and eigenvalues. Characteristic polynomial. Minimal polynomial. Diagonalization and triangularization.

8) Generalized eigenvectors. Spectral decomposition.

9) Jordan canonical form. Hamilton-Cayley Theorem.

10) Vectors and covectors. Dual vector spaces. Dual linear maps. Cokernel and coimage.

11) Inner products. Orthogonal complements.

12) Symmetric and Hermitian matrices. Orthogonal and unitary matrices. Diagonalization.

13) Positive definite matrices.

14) Tensor products of linear vector spaces.

15) Exterior powers of linear vector spaces.