Mathematics: Linear Algebra

<p>The course is a compulsory course for first-cycle studies for a Bachelor of Science degree in mathematics and in physics. The course can also be given as a stand-alone course or as part of a course package. The overarching goal of the course is for the students to acquire basic knowledge of linear algebra which is necessary for further studies in mathematics and natural sciences. Special emphasis is placed on developing the mathematical theory for vector spaces in a systematic way that contributes to strengthening the students' ability to absorb mathematical text, to conduct mathematical reasoning, to solve problems of both theoretical and applied nature and to communicate mathematics.</p><p>The course treats</p> <ul> <li>Matrices: matrix operations, matrix inverse, matrix rank</li> <li>Determinants: definition and properties</li> <li>Linear spaces: subspace, span, linear dependence/independence, basis, dimension</li> <li>Euclidean spaces: scalar product, Cauchy-Schwarz inequality, orthogonality, orthonormal bases, orthogonalisation, orthogonal matrices, orthogonal projection, orthogonal complement, least squares method, isometries.</li> <li>Linear mappings: matrices of linear mappings, kernel and image, composition and inverse, change of basis, rank-nullity theorem</li> <li>Spectral theory: eigenvalues, eigenvectors, eigenspace, characteristic polynomial, diagonalisability, the spectral theorem</li> <li>Systems of linear ordinary differential equations</li> <li>Quadratic forms: bilinear forms, diagonalisation, quadratic curves, quadratic surfaces, law of inertia</li> </ul>