Course syllabus - Vector Algebra

Objectives

The purpose of the course is to introduce the fundamental theory of systems of linear equations, vector algebra and matrices, and to show how it can be used as a tool of analysis in applications. Furthermore, the course aims to give a basis for further studies in mathematics and applications thereof in natural science, technology and economy.
 

Learning outcomes

At the end of a passed course, the student is expected to be able to …
- solve systems of linear equations by Gaussian elimination.
- apply and graphically illustrate the arithmetic operations for vectors in the plane, in 3-space and in Rn, and based on the concepts linear dependence/independence, basis, coordinates and change of basis, analyse and compare vectors.
- formulate and geometrically describe equations for straight lines and planes in 3-space, on parametric as well as non-parametric form.
- apply the scalar and vector products for evaluations of angles, lengths/distances, areas and volumes in geometrical applications that can be illustrated by vectors.
- account for and illustrate geometrically the fundamental properties of complex numbers, carry out arithmetic operations with complex numbers, pass between the rectangular form and the polar form of a complex number, solve binomic equations and second order equations of complex numbers, and apply the fundamental theorem of algebra to completely factorize polynomials with real coefficients.
- apply and explain the arithmetic operations and laws of matrix algebra, decide whether a quadratic matrix is invertible or not, calculate inverses and solve affine matrix equations.
- interpret matrices as linear transformations from Rn to Rm, find kernels and images of linear transformations, define rotations, reflections and orthogonal projections in the plane and in 3-space, and determine the matrices of such transformations.
- apply and describe the definition of the determinant and its interpretation as the volume of a parallelepiped in a n-dimensional space, and apply the laws of determinant arithmetics, particularly the multiplicative property and the expansion along a row or a column.
- explain and apply what is known as the fundamental theorem of quadratic matrices, which in different but equivalent terms expresses that the determinant of a quadratic matrix is non-zero if and only if the matrix is invertible.
- geometrically interpret and apply the concepts eigenvalue and eigenvector, find eigenvalues and eigenvectors of linear operators, solve elementary eigenvalue problems, and determine whether or not a given vector is an eigenvector of a linear operator and use this in applicable cases to perform a change of basis that diagonalizes a linear operator.

Course content

- Linear systems of equations: Gaussian elimination, row equivalence, row-echelon form, rank.
- Geometry in the plane and in 3-space: directed line segments, vectors, bases, coordinates, coordinate systems, lines and planes.
- Geometry in Rn: vectors, linear dependence/independence, linear transformations, kernel, image, the interpretation of a matrix as a linear transformation, matrices of rotation, reflection and orthogonal projection in R2 and R3.
- Scalar product: orthogonal projection, orthogonal basis, orthonormal basis, geometrical applications such as finding of mirror points, distances and angles.
- Vector product: right oriented orthonormal basis, geometrical applications.
- Complex numbers: arithmetical operations, rectangular form, polar form, binomial equations, second order equations over complex numbers, the factor theorem.
- Matrices: types of matrices, arithmetic operations and arithmetic laws, inverses, affine matrix equations, coefficient matrices, matrices of linear transformations, orthogonal matrices.
- Determinants: the product rule, determinants of transposes and inverses, arithmetic laws for determinants, expansion in minors along a row or a column.
- Eigenvalue problems: eigenvalue, eigenvector, characteristic polynomial, diagonalization.
 

Tuition

Teaching is given in the form of lectures and classes.
 

Requirements

Mathematics D or Mathematics 4.

Examination

Examination (TEN1), 3,5 credits, marks 3, 4 or 5
Examination (TEN2), 4,0 credits, marks 3, 4 or 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

Grading scale: 5, 4, 3