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Graduate courses

Departments' graduate courses for PhD-students.

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Course syllabus for

Academic year
MVE023 - Linear algebra
Linjär algebra
 
Course syllabus adopted 2021-02-05 by Head of Programme (or corresponding)
Owner: TKIEK
6,0 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: First-cycle
Main field of study: Mathematics
Department: 11 - MATHEMATICAL SCIENCES


Teaching language: Swedish
Application code: 51129
Open for exchange students: No
Only students with the course round in the programme overview
Status, available places (updated regularly): Yes

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0121 Examination 6,0 c Grading: TH   6,0 c    

In programs

TKIEK INDUSTRIAL ENGINEERING AND MANAGEMENT, Year 1 (compulsory)

Examiner:

Jan-Alve Svensson

  Go to Course Homepage


Eligibility

General entry requirements for bachelor's level (first cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Aim

The purpose of the course is to, together with the other mathematics
courses in the program, provide a general knowledge in the mathematics
required in further studies as well as in the future professional
career.

Learning outcomes (after completion of the course the student should be able to)

After completing the course the student shall

- Understand the idea behind systems of linear equations and their solutions, be able to rewrite them in matrix form and use them for basic modeling.
- Know what a matrix is and what the (elementary) row operations acting are and use them to transform a matrix to (reduced) echelon form to determine consistency of linear equations systems and describe the solution in a minimal parametrized form.
- Be able to account for the algebraic operations on matrices, the rules that applies to them, use them and understand their connections with row operations and linear systems of equations when solving problems.
- Be able to account for the concept determinant of a matrix, now its basic properties and theorems in this context, be able to prove some of them and use them when computing determinants and solving problems.
- Be able to account for when a matrix is invertible and in this case be able to determine the inverse, or specific entry in the inverse, using both row operations and determinants .
- Be able to account for the concepts vector space, subspace, linearly independent vectors in, basis for and dimension of such a space, know basic theorems in this context, be able to prove some of these, and determine and use them when solving problems.
- Be able to account for the concept of linear transformation between vector spaces, determine its matrix with respect to bases and be able to use this when solving problems.
- Be able to account for properties of linear transformations, their connection to matrices and systems of linear equations, and be able to determine and use them in problem solving.
- Understand and be able to determine (basis for) kernel and image of a linear transformation, null space and column space of a matrix, and be able to determine if/when a vector is in one of them.
- Understand the algebraic operations on linear transformations and the connection to such for matrices and be able to use this in problem solving.
- Understand and geometrically interpret the concepts eigenvector and eigen value of a linear operator, know the basic theorems in this context, be able to prove some of these, and be able to determine them and use then when solving problems.
- Be able to account for the concept diagonalizable matrix, know the basic theorems in connection with this and use them to diagonalize when possible.
- Be able to use diagonalization to solve discrete linear dynamical systems and systems of first order linear differential equations.
- Be able to account for the concepts inner product, orthogonality, orthogonal complement of a subspace and use this in problem solving.
- Be able to determine an orthogonal/orthonormal basis for a subspace, the matrix on orthogonal projection on a subspace, and use least squares to determine a best approximative solution and curve fitting.
- Be able to account for the concept quadratic form, determine the symmetric matrix for such a form, and use diagonalization and completion of the square to classify it.
- Show evidence of entrepreneurial skills by in a group of peers identify and make a conscious choice of idea related to the other learning outcomes that can benefit future students, and implement it using own resources and those of others.

Content

 - Solvability and solution of linear systems of equations using row operations on matrices.
- Matrix algebra, determinants, invertibility and invers of a matrix.
- Null space and column space of a matrix.
- Vector spaces and their subspaces, bases and dimensions.
- Linear transformation between vector spaces, matrices, kernel and image of such a transformation.
- Eigenvalues and eigenvectors of linear operators and matrices
- Diagonalization of linear operators.
- Discrete linear dynamical systems and systems of first order linear equations using diagonalization.
- Vector spaces with inner product, orthogonal and orthonormal bases and orthogonal projections.
- Least squares with applications to linear models.
- Quadratic forms and their classification.

Organisation

Instruction is given in lectures, classes and laboratory sessions. More detailed information
will be given on the course web page before start of the course.

Literature

 Literature will be announced on the course web page before start of the course.

Examination including compulsory elements

More detailed information about the examination will be given on the course web page before start of the course.
Examples of assessments are:
- selected exercises are to be presented to the teacher orally or in writing during the course,
- optional assessment during the course that can give bonus points,
- entrepreneurial project work assessed by delivery of result,
- written final exam after the course.
- exercises solved using software assessed by a teacher at the computer.


The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers on educational support due to disability.


Page manager Published: Thu 04 Feb 2021.