MATH-4100, LINEAR ALGEBRA

# LINEAR ALGEBRA

Time: Tuesday, Friday, 12:00 to 1:50 PM
Room: Darrin 337

Instructor: Gregor Kovacic
Office: 420 Amos Eaton
Phone: 276-6908
E-mail: kovacg at rpi dot edu

Teaching Assistant: Nour Al Hassaineh
Office: 433 Amos Eaton
E-mail: alhasn at rpi dot edu
Office Hours: Monday, Wednesday, Thursday, 10:00 to 11:00 AM

• This document contains the list of topics to be presented in this course, a list of textbooks that will be on reserve in the library, and some comments and recommendations about these textbooks.
• Click on each topic title to download the notes for that topic.
• To check out the grading and homework rules, click here.
• To see the test dates and topics, click here.
• To find a discussion of the homework assignments, click here.
• To find statements of attendance policy, academic integrity, how to compute your current grade, grade appeals, etc., click here.

## Topics

Click on each topic title to download the notes for that topic. You can access much nicer notes below.

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.

The daily build of class notes is deposited here.
Class notes from the Summer session of 2018 are deposited here.
Class notes from the Fall semester of 2017 are deposited here.
Class notes from the Fall semester of 2014 are deposited here.
Class notes from the Fall semester of 2012 are deposited here.

## Textbooks

The following textbooks contain material similar to that presented in this course:

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.