**Time:** Monday, Thursday, 10:00 to 11:50 AM

**Room:** Lally 102 **Instructor:** Gregor
Kovacic

**Office:** 419 Amos Eaton

**Phone:** 276-6908

**E-mail:** kovacg@rpi.edu

**Office Hours:** Click here.

**Vectors in the three-dimensional space:** Vectors and scalars, vector sum,
multiplication by a scalar, dot product, cross product, triple product, line and plane in space,
distance between points and lines and planes, sphere and cone. Orthogonal and non-orthogonal
coordinate systems, basis.

**Complex numbers:** Sum, product, absoulte value, representation in the plane,
complex conjugate, absolute value, polar representation, De Moivre's formula, roots of unity,
triangle inequality, complex exponential.

**Systems of linear equations:** Gausian elimination, overdetermined and
underdetermined systems.

**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.

**Determinants:** Projections and normals, volumes and oriented volumes in
*n*-dimensional spaces, algebraic
properties of determinants, minors and cofactors, multiplication of determinats,
calculation of inverse matrices, Cramer's rule.

**Eigenvalues and eigenvectors:** Characteristic polynomial, spectrum,
diagonalization, spectral theory of normal, self-adjoint, and unitary operators, simultaneous
diagonalization and commutativity, positive definite matrices and polar decomposition.

**Bilinear and quadratic forms:** Reduction of a quadratic form to a sum of
squares, law of inertia, diagonalizing a quadratic form using elementary row and column operations,
simultaneous diagonalization of matrices and quadratic forms.

**Singular-value decomposition:** Incompatible systems and the method of least squares, normal system,
singular value decomposition, low-rank approxmation of a matrix, quasi-inverse.

**Jordan normal form of a matrix:** Alegbraic and geometric multiplicity of an eigenvalue,
matrices that cannot be diagonalized, invariant subspaces and direct sums of operators,
nilpotent matrices, Jordan normal form,
minimal polynomial, Cayley-Hamilton theorem.

Class notes from the Fall semester of 2012 are deposited here.

T. M. Apostol, *Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications
to Differential Equations and Probability,* Wiley.

M. A. Akivis and V. V. Goldberg, *An Introduction to Linear Algebra and Tensors,* Dover.

R. Courant and F. John, *Introduction to Calculus and Analysis, Vol. II/1,* Springer.

C. G. Cullen, *Matrices and Linear Transformations,* Dover.

H. Dym, *Linear Algebra in Action,* AMS.

J. N. Franklin, *Matrix Theory,* Dover.

F. R. Gantmacher, *The Theory of Matrices,* Chelsea.

I. M. Gelfand, *Lectures on Linear Algebra,* Dover.

P. R. Halmos, *Finite-Dimensional Vector Spaces,* Springer Verlag.

L. Mirsky, *An Introduction to Linear Algebra,* Dover.

P. D. Lax, *Linear Algebra and its Applications,* Wiley.

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.

J. T. Scheick, *Linear Algebra with Applications,* McGraw-Hill.

G. E. Shilov, *Linear Algebra,* Dover.

G. Strang, *Introduction to Linear Algebra,* Wellesley Cambridge Press.

G. Strang, *Linear Algebra and Its Applications,* Brooks Cole.

If you want to read a book with a really nice, somewhat more matrix-oriented perspective than the course, read one of the books by Strang. They are very popular texbooks, written by a great expert in computational mathematics from a practical standpoint. He really explains the material well without any unnecessary abstraction. The books by Gelfand and Shilov are old classics from the Russian school. The book by Halmos is a U.S. classic, probably the first modern exposition of linear algebra. They are on a somewhat high level, especially the book by Halmos. If you have any intention to ever go to graduate school in mathematics, you should own at least the books by Gelfand and Halmos. The book by Lipschutz is a good reference and source of practical problems. The books by Akivis and Goldberg, Cullen, Franklin, Marcus and Minc, Mirsky, and Scheick are nice textbooks. The books by Dym and Lax are contemporary classics, rather advanced, and written at a very high level. The two books by Gantmacher are old monographs on matrix theory, full of results that cannot be found elsewhere. The books by Apostol and Courant and John are high-level calculus/introductory mathematical analysis books with good chapters on some aspects of linear algebra. I will use material from many of these books in my lectures. Finally, here is a nice book in electronic form.