Linear algebra and analytic geometry

Objectives and outcomes

Acquisition of general and expert knowledge of linear algebra and analytic geometry. Upon completion of the course, students will learn the basic concepts of linear algebra and analytic geometry, as well as the most important theorems covered in the course and will be able to apply the above to other courses.


Systems of linear equations. Gauss’s method. Gauss-Jordan reduction. Homogeneous systems. Matrices. Operations with matrices. Square matrices. Diagonal, triangular, symmetric matrices and invertible matrices. Rank of matrices. Kronecker-Capelli theorem. Application of matrices to solve linear systems. Determinants. Definition and properties. Laplace’s formula, Kramer’s theorem and matrix inversion. Vectors in real space of ordered tuples. Vector operations. Norm, angle and distance. Analytic geometry in R3 – lines and planes. Vector spaces – axioms, subspaces, intersection, sum. Grassman’s formula. Linear independence. Spans. Spanning sets. Row and column spaces of matrices. Zero space. Basis and dimension. Linear maps. Kernel and image of linear maps (range space and null space). Representing linear maps with matrices. Change of basis. Similarity. Diagonalization of linear operators. Eigenvalues and eigenvectors. Characteristic and minimal polynomial of square matrix. Cayley-Hamilton theorem. Gram-Schmidt orthogonalization. Orthogonal projection into a line. Projection into a subspace. Orthogonal and unitary matrices. Conics and quadratics surfaces. The canonical equations of curves and surfaces of the second order.

Practical classes

Solving tasks from the units covered in the lectures.