SF1624 Algebra and Geometry 7.5 credits

• Educational level

  First cycle

• Academic level (A-D)

  A

• Subject area

  Mathematics
  Techonology
  

• Grade scale

  A, B, C, D, E, FX, F

• Link to homepage

• Course page on KTH Social

Learning outcomes

After completing the course students should for a passing grade be able to

• use the basic concepts and problem solving methods in linear algebra and geometry. In particular it means to be able to:
  - understand, interpret and use the basic concepts: the vector space Rn, subspaces of Rn, linear dependence and independence, basis, dimension, linear transformations, matrix, determinant, eigenvalue and eigenvector.
  - solve geometric problems in two and three dimensions using for example vectors, dot product, vector product, triple product and projection.
  - use Gauss-Jordan?s method for example to solve linear systems of equations, calculate inverse matrices, determinants and to resolve questions about linearly independent.
  - use matrix and determinant calculus to address issues regarding linear transformations and linear systems.
  - use the least-squares method to solve for example problems with over-determined linear systems of equations.
  - use different bases for vector spaces to handle vectors and linear transformations, and to manage changes of bases and linear coordinate transformations.
  - compute eigenvalues and eigenvectors and use this for example in order to diagonalize matrices, to study quadratic forms, conics in the plane and quadratic surfaces in three space.
  - use the Euclidean inner product in order to address the questions 
  about distance, orthogonality and projection, and apply Gram-Schmidt?s 
  method to calculate orthogonal bases of subspaces.
• set up simple mathematical models where the fundamental concepts in linear algebra and geometry are used, discuss the relevance of such 
  models, reasonableness and accuracy, and know how mathematical  software can be used for calculations and visualization.
• read and understand mathematical texts about for example,  vectors, matrices, linear transformations and their applications, communicate mathematical reasoning and calculations in this area, orally and in writing in such a way that they are easy to follow.

For higher grades, the student in addition should be able to:

• manage general vector spaces, such as function spaces or vector spaces of matrices.
• use other inner products than the Euclidean inner product.
• derive important relations in linear algebra and geometry.
• generalize and adapt the methods to use in somewhat new contexts.
• solve problems that require synthesis of material and ideas from all over the course.
• describe the theory behind concepts such as eigenvalues and orthogonality.

Course main content

Vectors, matrices, linear equations, Gaussian elimination, vector  geometry with dot product and vector product, determinants, vector spaces, linear independence, bases, change of basis, the least-squares method, eigenvalues, eigenvectors, quadratic forms, orthogonality, inner-product space, Gram-Schmidt?s method

Eligibility

Basic and specific requirements for engineering program.
Mandatory for first year, can not be read by other students

Literature

Examination

• TEN1 - Examination, 7.5 credits, grade scale: A, B, C, D, E, FX, F

Requirements for final grade

Written exam, possibly with the possibility of continuous examination.

Offered by

SCI/Mathematics

Examiner

Roy M Skjelnes <skjelnes@kth.se>

Version

Course plan valid from: Autumn 10.
Examination information valid from: Autumn 07.