MATH 0540, Fall 2009. Honors Linear Algebra

MWF 10:00--10:50 am,  Barus & Holley 158

Instructor: Prof. Sergei Treil,
office: Kassar 216,   ph.: x1122
e-mail: treil<at>math.brown.edu
Web: http://www.mth.brown.edu/~treil/

Office hours: MWF 2:00-3:00 pm and by appointment.

Office hours for the exam week: Fri 12/12 10:00am--1:30 pm, and Mon. 12/15, 10:30am--12:00 noon

Book: Linear Algebra done Wrong, by S. Treil

Syllabus.(pdf format)

Outline of the course and homework assignments.

New: solutions to the first HW assignments and to the problem # 6.10 (3d rotation) is posted on "myCourses"

Lecture Notes taken by students (pdf format). 

• 9/9. Vector spaces, 56K
• 9/11. Linear combinations. Basis. Generating and linearly independednt systems. (68K)     
• 9/14. Linear transformations. Matrix of a linear transformation. Matrix-vector multiplication.  (56K) 
• 9/16. Composition of linear transformations and matrix multiplication. (60K) 
• 91/8. Invertible transformations and matrices.  (108K) 
• 9/21: Subspaces. Review of invertible matrices.  (116K)
• 9/23: System of linear equations. Echelon reduction.  (64K)
• 9/25: Echelon and reduced echelon form. Analyzing the pivots. (44K) 
• 9/28: Introduction to proofs. Finding inverses by row reduction. Bsisis, dimension.  (44K)
• 9/30: Structure of the general solution of a linear system.  
• 10/02: Computimg bases in fundamental subspaces. Rank. Rank theorem. Completing LI systems to bases.  
• 10/05: Change of coordinates formula.  
• 10/07: Introduction to determinants.    
• 10/09: Properties of determinant.  
• 10/14: Review for the midterm.  
• 10/19: Formal definition of determinant. ermutations.  
• 10/21: Cofactor expansion of determinant. Cofactr formula for he inverse. (Kramer's rule and  minors and rank --- read yourself).  
• 10/23: Proof of cofactor formula for inverse. Eigenvectors and eigenvalues: main definitions and examples. How to compute eigenvalues and eigenvectors.  
• 10/26: Diagonalization.  
• 10/28: Diagonalization (continued).  
• 10/30: Inner product spaces. Main definitions. Cauchy-Schwarz inequality.  
• 11/02: Norms. Norms generated by inner prducts. Polarization identity and parallelogramm equality. Orthogonal and orthonormal bases. (Baby version of ) Abstract Orthogonal Fourier decomposition.  
• 10/04: Orthogonal projections and Gram--Scmidt orthogonalization.    
• 11/06: Least square solution. Another formula for orthogonal projection.    
• 11/09: Adjoint matrix and adjoint transformation. Relation between funcdamental subspaces.  
• 11/11:  Rigid motions, isometries and unitary transformations.    
• 11/13: Upper triangular (Schur) representation. Spectral Theorem for self-adjoint and normal transformations.
• 11/16: Polar and singular value decompositions.  
• 11/18: Singular value decomposition (continued).          
• 11/20: Review for the exam.