Subject Area  Applications and Foundations of Computer Science 

Semester  Semester 1 – Fall 
Type  Required 
Teaching Hours  4 
ECTS  6 
Course Director 
Emmanouil Vavalis, Professor 
Course Instructor 

 Basics of Linear Algebra.
 Introduction, vectors, matrices, operations with matrices, properties.
 Linear algebraic systems, Gauss Elimination, analysis LU, analysisCholesky, complexity analysis of Gauss elimination.
 Inverse matrices.
 Rectangular matrices.
 Linear independence and vector spaces.
 Bases.
 Basic vector spaces of a matrix, existence and uniqueness of the solution. Inner product, distances, norms andvector orthogonality.
 Quadratic matrices and least squares method.
 Symmetric and positive definite matrices.Linear transformations.
 Determinants and their properties.
 Eigenvalues and eigenvectors, orthonormality, singular value decomposition, Jordan’s normal form.
The main objective of the course is the deep understanding of the fundamental concepts of Linear Algebra which the Department (and almost all other similar departments around the world) considers necessary for the continuation of studies. Sub objective of this course is to acquire specific technical knowledge (e.g. how can I solve a linear system, how can I calculate the eigenvalues, …) but which by themselves would be completely useless.
In particular students will be capable to:
 Demonstrate competence with the basic ideas of linear algebra including concepts of linear systems, independence, linear transformations, bases and dimension, eigenvalues, eigenvectors and diagonalization.
 Compose clear and accurate proofs using the concepts of this course.
More specifically they will be able to:
 Determine if a system of equations is consistent and find its general solution.
 Row reduce a matrix to reduced echelon form.
 Apply solution methods of linear system for various problems.
 Solve the equation Ax = b where A is an m x n matrix and x is in ℜ^n
 Write the solution set of a given homogeneous system in parametric vector form.
 Determine if the columns of a given matrix form a linearly dependent set.
 Understand the linear transformation defined by x → Ax.
 If T is a linear transformation, find the standard matrix of T.
 Matrix algebra including the inverse of a matrix.
 Recognize various characterizations of nonsingular matrices.
 Compute the products of matrices, which are partitioned conformably.
 Compute the determinant of a given matrix.
 Combine row reduction and cofactor expansion to compute a given determinant.
 Determine a subspace from a vector space.
 Determine a null space and a column space.
 Find bases for vector spaces.
 Find the dimension of the subspace spanned by the given vectors.
 Given an m x n matrix, find the rank and nullity of the matrix.
 Change the coordinates of a vector from a basis to a standard basis.
 Find the characteristic polynomial of a given matrix.
 Given a matrix and an eigenvalue, find the basis for the corresponding eigenspace.