Instructor: Alexander Borisov
Office: 414 Thackeray Hall
Office hours: MW 12-1pm + by appointment
MWF 10:00-10:50 am in 704 Thackeray Hall.
Th 10:00-10:50 am in 704 Thackeray Hall. Recitations Instructor: Takuya Murata.
Final Examination date is set: Friday, December 9th. Format of the exam: one theorem, two problems. All topics are fair game, except 37 and 38. the problems may be harder that in the Fall 2010 final examination.
Homework due Monday, Nov. 21.
Midterm Examination date is set: Monday, October 24. Please see the Midterm Study Guide for more information.
A Day-by-day List of Topics covered so far may help you to review the material.
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.
Any linear algebra textbook that you can read, which is sufficiently advanced. It must include the notion of a quotient space, and the proof (not just the statement) of the Jordan Canonical Form Theorem. I also recommend the free online book Linear Algebra Done Wrong by Sergei Treil . This book will be used for some homework assignments.
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 may be assigned, 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 are subject to change. Some of the topics will take several classes to cover.
1) Definition of a linear vector space, subspace.
2) Linear maps between spaces; Ker and Im; Hom(V,W). Dual space, dual map.
3) Construction of the quotient space; first isomorphism and lattice isomorphism theorems.
4) Definition of the dimension (independence of basis);
5) Extension of a basis of a subspace to a basis of the space. Dimension of proper subspace.
6) Dimension of a sum of two subspaces. Complements.
7) Dimension and basis of Hom(V,W). Dual basis. Double dual theorem.
8) Annihilator and its properties.
9) Matrix of a linear map, change of basis formula;
10) Definition of the determinant; properties of the determinant.
11) Characterizations of invertible maps.
12) Invariant subspace. Map on the quotient. Matrix of an operator in a basis that extends a basis of an invariant subspace.
13) One-dimensional invariant subspaces, eigenvectors, eigenvalues. The eigenvalues are the roots of the characteristic polynomial.
14) Linear independence of eigenvectors with different eigenvalues.
15) Diagonalization. Characterizations of diagonalizable operators and matrices as having a basis of eigenvectors.
16) Minimal polynomial. Roots=eigenvalues theorem. Diagonalzation criterion.
17) Projections and complements; reflections.
18) Commuting maps and simultaneous diagonalization.
19) Generalized eigenspaces. Nilpotent operators and matrices; theorem that L^(dim V)=0 for a nilpotent L on V.
20) Linearly independent subspaces. Linear independence of the generalized eigenspaces.
21) Partial Fractions Theorem.
22) Spectral decomposition (V is the direct sum of the generalized eigenspaces for L).
23) Algebraic multiplicity is the dimension of the generalized eigenspace.
24) Hamilton-Cayley Theorem.
25) The existence of the Jordan basis for a nilpotent map. Jordan Canonical Form Theorem.
26) Jordan canonical form and power series on matrices.
26) Every operator can be uniquely written as the sum of commuting semisimple and nilpotent.
27) Bilinear forms, matrix of a form.
28) Degenerate and non-degenerate forms, symmetric forms, positive-definite forms and matrices. Change of basis formula.
29) Quadratic functions are in 1-to-1 correspondence with the symmetric bilinear forms.
30) Positive-definite forms and distances; triangle inequality and Cauchy-Schwarz inequality; Euclidean vector spaces; identification of dual with the original space.
31) Orthonormal basis; Gram-Schmidt algorithm; every finite-dimensional Euclidian space is isomorphic to R^n with the dot product.
32) Adjoint of an operator from one Euclidean space to another.
33) Self-adjoint operators and (real) symmetric matrices; orthogonal matrices.
34) Orthonormal diagonalization of symmetric matrices.
35) Diagonalization of symmetric bilinear forms; definition of signature (theorem that it is well defined).
36) Sylvester's characterization of positive-definite matrices.
37) Tensor products of linear vector spaces.
38) Exterior powers of linear vector spaces.