Linear Algebra

Aim

The student will learn basic mathematical concepts and methods in linear algebra that are used by statisticians. The course objective is for the student to be able to read and understand linear algebra in scientific statistical texts, and to be able to conduct logical reasoning and linear algebra calculations. This includes that the student will - be able to solve systems of linear equations. - know the concept of a vector in arbitrary dimenstion. - be able to calculate scalar products and projections of vectors. - know the concepts of a matrix and be able to perform matrix calculations. - be able to calculate determinants and know what the determinant say about linear dependencies. - know examples of linear maps and how to represent these by matrices. - know the concepts of basis and coordinates, and be able to use orthogonal matrices for change of bases. - be able to determine eigenvalues and eigenvectors. - be able to diagonalize symmetric matrices and use this on quadratic forms. - be able to use the method of least squares and know about the geometrical interpretation.

Content

The following be treated in the course. - Linear systems of equations: succesive elimination and substitution, possible solutions, geometrical interpretation. - Matrices: multiplication, transpose, rank, trace, inverse, easy equations. - Vectors: geometrical vectors, scalar product, projections, coordinates, linear combinations, linear independencies/dependencies. - Bases: orthonormal bases, change of bases, orthogonal matrices, Gram-Schmidt process. - Determinants: definition, calculation of orders 2 and 3, relation to linear dependencies and systems of equations. - Linear maps: geometrical examples, matrix representation. - Diagonalization: eigenvalues, eigenvectors, spectral decomposition, calculations for matrices of order 2 and 3. - Quadratic forms: matrix representation, diagonalization. - Method of least squares: overdetermined systems of equations, geometrical interpretation, curve-fitting.