Time: Monday, Wednesday, 2:00 to 3:50 PM
Room: Carnegie 112
Instructor: Gregor
Kovacic
Office: 419 Amos Eaton
Phone: 276-6908
E-mail:
Topics
Vector spaces: Vectors and scalars, linear combinations, linear
independence, basis, linear subspaces, direct sums.
Euclidean and unitary spaces: Inner product, Cauchy-Schwartz and triangle inequalites, distance in n-dimensional spaces, angle between two vectors, orthogonality, orthogonal bases, Gram-Schmidt orthogonalization, orthogonal complements.
Linear operators and matrices:Matrix of a linear operator in a given basis, algebra of linear operators, change of basis, adjoint operators, range and kernel, rank and nullity, Fredholm alternative, linear functionals, self-adjoint, orthogonal, unitary, and positive definite operators, invariant subspaces and direct sums of operators, projectors.
Determinants: Oriented volumes in n-dimensional spaces, algebraic properties of determinants, minors and cofactors, multiplication of determinats.
Systems of linear equations: Gausian elimination, overdetermined and underdetermined systems, Cramer's rule, calculation of inverse matrices, incompatible systems and the method of least squares.
Bilinear and quadratic forms: Reduction of a quadratic form to a sum of squares, law of inertia.
Eigenvalues and eigenvectors: characteristic polynomial, spectrum, diagonalization, spectral theory of normal, self-adjoint, and unitary operators, simultaneous diagonalization of matrices and quadratic forms, simultaneous diagonalization and commutativity, minimax principles.
Decomposition and normal forms of matrices: Polar decomposition, singular-value decomposition, nilpotent matrices, Jordan normal form, minimal polynomial, Cayley-Hamilton theorem.
Applications in differential equations and mechanics: Homogeneous linear systems of differential equations with constant coefficients and their general solutions, small oscillations and Lagrangian mechanics.
Textbooks
The following textbooks have been ordered as optional reading for this course
and are for sale in the Campus Bookstore:
I. M. Gelfand, Lectures on Linear Algebra, Dover.
P. R. Halmos, Finite-Dimensional Vector Spaces, Springer Verlag.
S. Lipschutz, Schaum's Outline of Theory and Problems of Linear Algebra,
Schaum's Outline Series, McGraw-Hill.
M. Marcus and H. Minc, Introduction to Linear Algebra, Dover.
G.E. Shilov, Linear Algebra, Dover.
The books by Gelfand, Halmos, and Shilov are true classics. They are on a somewhat high level, especially the book by Halmos. The book by Lipschutz is a good reference and source of practical problems. The book by Marcus and Minc is a nice textbook. If you have any intention to ever go to graduate school in mathematics, you should own at least the books by Gelfand and Halmos. I will use material from all these books in my lectures.