Linear Algebra and Geometry

The course takes up basics of logic and set theory, methods of proof, and complex numbers. We solve linear equations using Gaussian elimination, and learn to write equations with vectors and matrices, and to interpret row operations as multiplication with elementary matrices. We discuss matrix calculus in general, including finding the inverse of a matrix arithmetic rules inverses, transposed, and the like. Geometrically we begin studying properties of vectors in the plane and space (including dot product, cross product). From there, we develop the concepts of subspaces, basis, dimension, and abstract vector spaces. Special emphasis is placed on the vector spaces attached a matrix (null space, column space, row space) and the rank-nullity theorem. We consider linear maps, both geometrically and algebraically, and show how the matrices describing a linear map changes when changing the bases. Determinants are introduced, both as a criterion for when matrices are invertible, and in dimension 2 and 3 as area and volume. We show Cramer's rule. Eigenvalues and vectors are introduced. It is proven that a matrix is diagonalisable if and only if there exists a basis consisting of eigenvectors. We show that real symmetric matrices always are orthogonally diagonalizable, and uses this in the principal axis transformation to investigate / classify conic sections.

1. Knowledge. The student knows the basic concepts and methods in linear algebra, including vector spaces, subspaces, basis, dimension. Moreover, students know linear maps, both algebraically / in matrix form (including solution of linear systems of equations) and geometrically (including eigenvalues and eigenvectors). 2. Skills. The student is able to recognize linear problems and formulate them using linear equations and solve them using matrices and Gaussian elimination. The student is able to work with linear maps using matrices, including on geometric problems. In particular, the student is able to study conic sections using principal axis transformations. The student is able to give elementary mathematical proofs and do calculations using complex numbers.

Lectures, exercises and written final examination. Parts of this course can be taught in English.