Mathematics 2D: Linear algebra and differential equations

Complex numbers.

Linear systems of equations. Gaussian elimination with pivoting. Existence, uniqueness, and parametrization of solutions.

Vector equations. Linear span. Matrices. LU factorization.

Linear independence and basis. Determinants. Vector spaces and subspaces. Row space, column space, and null space of a matrix. Rank and nullity. Linear transformation and change of basis.

Vector and matrix norms. Error analysis. Condition numbers.

Projection of a vector onto a subspace. Orthogonal projection. Inner product. Outer product. Orthogonal basis. Gram-Schmidt method. Householder transformations and QR factorization.

Eigenvalues. Eigenvectors. Diagonalization. Defective matrices.

Singular value decomposition. Systems of linear differential equations. Matrix exponential. Second-order linear differential equations.

Examples of mathematical modeling and applications in the natural sciences and technology.

Programming algorithms in linear algebra.

The student understands and can apply basic concepts, results, and methods from linear algebra concerning solving systems of linear equations and systems of first-order differential equations. The student has knowledge of algorithmic thinking in order to understand and apply basic numerical methods for solving systems of linear equations and for solving systems of first-order differential equations. Furthermore, the student can analyze such methods with regard to applicability and precision.

The student is familiar with the use of numerical methods in a programming language and understands the possibilities and limitations of the various methods in relation to the problems they are applied to.

The student can use analytical and computational methods to formulate, model and solve simple technological problems relevant to their study programme.

The course will primarily contribute to competence area K1; show specialist knowledge and a professionally grounded perspective. It will also contribute to competence area K2; analysing engineering problems, in collaboration with the various study programmes that the subject serves.

Lectures and compulsory exercises. The number of exercises that must be approved will be stated at the start of the semester on the course's website. The course will be taught in Norwegian.