Linear algebra and analytic geometry

Vectors in real and complex space of ordered tuples; norm, angle and distance. Systems of linear equations. Gaussian process. Homogeneous systems. Matrices. Operations with matrices, application to solving linear systems. Algebra of square matrices. Diagonal, triangular, symmetric matrices and invertible matrices. Vector spaces, axioms, a direct product of spaces; vector subspaces, intersection and sum. Space of types of matrices. Base and dimension. Grasmane formula. Coordinates. Rank of matrices and linear systems. Linear mappings. Kernel and image, the application to linear systems. Algebra of linear operators. Matrices and linear mappings, change of base, similarity.

Determinants. Definition and properties. Development, Kramer's theorem and matrix inversion. Diagonalization of linear operators. Eigenvalues and eigenvectors. Characteristic and minimal polynomial  of square matrix.. Diagonalization. Cayley-Hamilton theorem. Bilinear and quadratic forms. Diagonalization.

The classification of complex and real symmetric forms. Vector spaces with scalar product. Norm, distance, angle. Gram-Schmitt orthogonalization, orthogonal projection, the distance between vector subspaces. Orthogonal and unitary matrices. Symmetric and Hermitian operators, diagonalization. Orthogonal and unitary operators, canonical bases and matrices. Applications in geometry. Affine spaces and subspaces. Solving geometric tasks using analytical methods.

Orthogonal projection and distance of a point from the subspace. Conics and quadratics surfaces. The canonical equations of curves and surfaces of the second order.

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