Linear Algebra MATH-3215

COURSE MATERIAL:

INTRODUCTION TO COURSE

The objective of this course is to provide fundamental of solution for system of linear equations, operations on system of equations, Matrix properties, solutions and study of their properties.

COURSE PRE-REQUISITE

None

COURSE CONTENTS

Introduction to vectors, vectors and linear combinations, lengths and dot products, matrices.
Solving linear equations, the idea of elimination, elimination using matrices, rules for matrix operations, inverse matrices.
Elimination, factorization A=LU, transposes and permutations.
Vector space and subspaces, space of vectors, the Null space of A, solving AX=0, the rank and the row reduced form,independance, basis and dimension,dimension of the four subspaces.
Orthogonality, projections, least square approximations, orthogonal basis and Gram Schmidt.
Determinents, properties of determinents, permutations and cofectors, Cramers rule, inverses and volumes.
Eigen values and Eigen vectors, diagonalizing a matrix, symmetric matrices, positive definite matrices, similar matrices, Singular Value Decomposition.
Applications of matrices in Engineering, graphs and networks, Markov matrices, population and economics, Linear Algebra for functions, Statistics and Probability, Computer Graphics.
Numerical Linear Algebra, Norms and condition numbers, Iterative methods for Linear Algebra.
Complex vectors and matrices, Hermitian and Unitary matrices, Matrix factorization.

LEARNING OUTCOMES

Basic learning outcomes is to provide students an understanding of:

Interpret existence and uniqueness of solutions geometrically.
Understand algebraic and geometric representations of vectors in R^n and their operations, including addition, scalar multiplication and dot product.
Recognize echelon forms.
Define the inverse of a matrix.
Compute the inverse of a matrix.
Provide a definition of the determinant.
Use determinants and their interpretation as volumes and applications.
Provide an axiomatic description of an abstract vector space.
Determine a basis and the dimension of a finite-dimensional space.
Find the eigenvalues and eigenvectors of a matrix.
Understand how to determine the angle between vectors and the orthogonality of vectors.
Define orthogonal projections.
Explain the Gram-Schmidt orthogonalization process
Applications of LINEAR ALGEBRA in different fields.