Course syllabus - Linear Algebra

Objectives

The course aims at giving a fundamental understanding of finite-dimensional linear spaces over real numbers and linear transformations between such spaces. The course also aims at giving a basis for further studies in mathematics and applications thereof in natural science and technology.

Learning outcomes

At the end of a passed course, the student is expected to be able to
1. define the meaning of a linear space over real numbers, be able to give examples of such spaces, and be able to determine whether a given set with given operations is a linear space or not.
2. for a finite set of vectors determine the subsets which are linear independent, and by that be able to find the dimension of a finite linear span.
3. find bases in finite-dimensional linear spaces, and be able to find the connection between the coordinates of a vector in two different bases.
4. define the meaning of a linear transformation and in a given basis be able to find its matrix, and also be able to explain the geometrical meanings of the general properties of linear transformations. Especially, the kernel and the range of a linear transformation should be able to be find and interpreted.
5. find the connection between the matrices of a linear operator in two different bases.
6. construct orthonormal bases in Euclidian spaces, and be able to project vectors orthogonally on subspaces of Euclidian spaces.
7. explain and apply the concepts of eigenvalue and eigenvector of a linear operator. Especially, a student is expected to be able to find the eigenspace belonging to an eigenvalue, and to be able to diagonalize a linear operator if possible.
8. apply the spectral theorem on symmetric linear operators.
9. diagonalize and classify quadratic forms on finite-dimensional linear spaces, and also in the plane and in the 3-dimensional space be able to interpret geometrically equations of quadrics.

Course content

- Linear space: definition of linear space over the real numbers, subspace, span, linear independence, dimension, basis, coordinates, change of basis, isomorphism.
- Linear transformation: definition of a linear transformation, linear operator, matrix representation, composition, inverse transformation, kernel, range.
- Euclidian space: Euclidian inner product, Euclidian space, orthogonality, orthogonal complement, ON-basis, orthogonal projection, Gram-Schmidt orthonormalization process, orthogonal matrix, isometric transformation.
- Spectral theory: eigenvalue, eigenvector, characteristic polynomial, eigenspace, diagonalizability, symmetric linear transformation, spectral theorem.
- Quadratic forms: definition of a quadratic form, diagonalization of a quadratic form, Sylvester’s theorem, rank, signature, positive definite, negative definite, semidefinite, indefinite, quadrics in the plane and in the 3-dimensional space.

Tuition

Teaching is given in the form of lectures and classes.

Specific requirements

Basic Vector Algebra, 7.5 credits, of which 2.5 credits must be completed at the beginning of the course, or equivalent.

Examination

INL1, Assigned problems, 2,5 credits, Assigned problems regarding learning outcomes 1-9, marks Fail (U), Pass (G).
TEN1, Examination, 5 credits, Written and/or oral examination regarding learning outcomes 1-9, marks Fail (U), 3, 4, 5.

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Pass with distinction, Pass with credit, Pass, Fail