## Linear Algebra, Math 405 Fall 2008

### Course Information

 Lecture 4122 CSIC Bldg. #406; TuTh 2-3:15pm Note special place: Math Bldg. Rm. 0305 on Tue 9/23 Note special place: Math Bldg. Rm. 0103 on Tue 11/4 Instructor Professor Eitan Tadmor Contact tel.: x5-0648   Email: eitan Tadmor Office Hours By appointment 4119 CSIC Bldg. #406 tel.: x5-0652   Email: eitan Tadmor Midterm Tuesday, Oct 14, 2-3:15pm 4122 CSIC Bldg. #406 (w/open books & open notes) Final Thursday, Dec 18, 10:30-12:30 4122 CSIC Bldg. #406(w/open books & open notes) Grading 30% Homework, 30% Midterm, 40% Final Grader Poorani Subramanian Email: poorani Subramanian

• PREREQUISITES: MATH 240  or MATH 461

• Course Description: This course contains an abstract treatment of finite dimensional vector spaces, linear transformations and their invariants.

• Contents

Vector spaces
Vector spaces and subspaces
Linear combinations, span and linear (in-)dependence

Note: On the transitivity of linear spans [ pdf file ]
##### Assignment #1 [ pdf file ] ... with answers [ pdf file]
Bases and dimension
Row equivalence in RN and rank
Coordinates

Note: An exchange theorem and the notion of dimension [ pdf file ]
##### Assignment #2 [ pdf file ] ... with answers [ pdf file]
Linear transformations
Linear transformations (homomorphism, isomorphism, ...)
Vector spaces of finite and infinite type; Rank and nullity, Sylvester theorem
(*) Direct sums and products of vector spaces
##### Assignment #3 [ pdf file ] ... with answers [ pdf file]
Matrix representation of linear transformations
Sum and product of matrices; change of bases

Note: On matrix representation of linear transformations [ pdf file ]
##### Assignment #4 [ pdf file ] ... with answers [ pdf file]
Linear functionals and duality
##### Assignment #5 [ pdf file ] ... with answers [ pdf file]        Midterm [ pdf file ] ... with answers [ pdf file]
Solution of linear systems of equations
Homogeneous and inhomogeneous systems
Determinants: definition and properties; Permutations and evaluation
Elementary matrix operations
Invertible matrices; Cramer's rule
##### Assignment #6 [ pdf file ] ... with answers [ pdf file]
Inner Product Spaces
Inner product
Orthonormal Bases; Gram-Schmidt; direct decompositions
Bessel, Parseval and Cauchy-Schwartz (in-)equalities
##### Assignment #7  [ pdf file ] ... with answers [ pdf file]
Positive, unitary and normal operators
The spectral theorem

Note:  On the the orthogonal decomposition of Hilbert spaces [ pdf file ]
##### Assignment #8 [ pdf file ] ... with answers [ pdf file]
Invariant decompositions
Direct sum decomposition's
Invariant subspaces
Eigenvectors and eigenvalues
##### Assignment #9 [ pdf file ] ... with answers [ pdf file]
Polynomials: the algorithm of Euclid; Factorization and (multiple) roots
Characteristic polynomial
Triangulation of matrices
Diagonalizable operators
Jordan canonical forms
##### Assignment #10 [ pdf file ] ... with answers [ pdf file]
Cayley-Hamilton: characteristic and minimal polynomials; companion matrix
Applications

Note:  Normal matrices are unitarily diagonalizable [ pdf file ]
##### Assignment #11 [ pdf file ] ... with answers [ pdf file]
Bilinear forms
General forms
Symmetric forms
Sylvester theorem
Principle coordinates

Note:  Bilinear forms, quadratic forms and congruence [ pdf file ]
##### Final ... [ pdf file ] ... with answers [ pdf file]
Euclidean geometry
3D vectors; scalar and vector products
Lines and relation between lines
Planes: definition and characterizations
Planes and lines
Change of coordinates; translation, rotation and reflection

### Epilogue - The sin(x) subroutine in MATLAB

References (Partial List)

• Linear Algebra, 2nd Edition, by Hoffman & Kunze,  Published by Prentice Hall. ISBN: 0135367972
• Linear Algebra, 3rd Edition, by S. Lang,  Published by Springer-Verlag.  ISBN: 0387964126
• Linear Algebra in Action by Harry Dym,  Graduate Studies in Math v. 78, American Math Society, 2007
• Linear Algebra, by Peter Lax
• Applied Linear Algebra, by Gilbert Strang
• Lecture Notes