1. Algebraic structures (groupoid, group, ring, intergral domain, skew field, field) 2. Matrices, basic operations, column space, row space of matrix, rang of matrix. 3. Determinant of matrix. Application. 4. Vector space, construction, linear independence of vectors, basis, dimension, subspaces of vector space, structure of subspaces. Examples of vector spaces. 5. Homomorphisms and isomorphisms of vector spaces. 6. Scalar product in vector space: norm and angle of vetors, orthogonality of vectors and subspaces, Gramm-Schmidt method of orthogonalisation. Isometry of vector spaces. 7. Systems of linear equations, solvability, solving methods. Gauss and Jordan methods. 8. Inverse and generalised inverse (Moor-Penrose) of matrices, connection with solving of systems of linear equations. Orthogonal and idempotent matrices, connection with projections of vector spaces. 9. Eigenvalue and eigenvectors of matrices, geometric interpretation. 10. Real symmetric matrices, positive and negative (semi)definite matrices, connection with eigenvalues and traces of matrices, spectral decomposition of matrices.
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