# Syllabus for Linear Algebra II

Linjär algebra II

A revised version of the syllabus is available.

## Syllabus

• 5 credits
• Course code: 1MA024
• Education cycle: First cycle
• Main field(s) of study and in-depth level: Mathematics G1F
• Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
• Established: 2007-03-15
• Established by: The Faculty Board of Science and Technology
• Revised: 2010-04-27
• Revised by: The Faculty Board of Science and Technology
• Applies from: week 35, 2010
• Entry requirements: Linear Algebra and Geometry I, Single Variable Calculus
• Responsible department: Department of Mathematics

## Learning outcomes

In order to pass the course (grade 3) the student should

• be able to give an account of and use basic vector space concepts such as linear space, linear dependence, basis, dimension, linear transformation;

• be able to give an account of and use basic concepts in the theory of finite dimensional Euclidean spaces;

• be able to compute determinants of arbitrary order;

• be familiar with the concepts of eigenvalue, eigenspace and eigenvector and know how to compute these objects;

• know the spectral theorem for symmetric operators and know how to diagonalise quadratic forms in ON-bases;

• know how to solve a system of linear differential equations with constant coefficients;

• be able to formulate important results and theorems covered by the course;

• be able to use the theory, methods and techniques of the course to solve mathematical problems;

• be able to present mathematical arguments to others.
• ## Content

Linear spaces: subspaces, linear span, linear dependence, basis, dimension, change of bases. Matrices: rank, column space and row space. Determinants of arbitrary order. Linear transformations: their matrix and its dependence on the bases, composition and inverse, range and nullspace, the dimension theorem. Euclidean spaces: inner product, the Cauchy-Schwarz inequality, orthogonality, ON-basis, orthogonalisation, orthogonal projection, isometry. Quadratic forms: diagonalisation, Sylvester's law of inertia. Spectral theory: eigenvalues, eigenvectors, eigenspaces, characteristic polynomial, diagonalisability, the spectral theorem, second degree surfaces. Systems of linear ordinary differential equations.

## Instruction

Lectures and problem solving sessions.

## Assessment

Written examination at the end of the course. Moreover, compulsory assignments may be given during the course.