Linear algebra, analytical and differential geometry

1. Matrix: definition, operations and properties. 2. The determinant of a matrix. Reverse matrix. The rank of a matrix. 3. Systems of linear equations. Cramer method. 4. Compatibility of systems of linear equations. Partial elimination method (Gauss). Total elimination method (Gauss-Jordan). 5. Laws of composition. Algebraic structures with laws of internal composition. 6. Vector spaces. Linear dependence and linear independence. 7. Generator system. The base of a vector field. The dimension of a vector space. 8. Real vector spaces with scalar product. Orthogonality. 9. Linear applications. The kernel and image of a linear application. 10. Lines in a plane. 11. Conics. Circle, ellipse, parabola, hyperbola. 12. Coordinate systems in space. The plan. Lines in space. 13. Curves in a plan. Tangent and normal of a plane curve. Curvature of a plane curve. 14. Curves in space. The tangent plane and the normal plane at a curve in space. Curvature and torsion of a curve in space.

Upon successful completion of this course students will be able to: 1) Use computational techniques and algebraic skills essential for the study of systems of linear equations, matrix algebra, vector spaces, eigenvalues and eigenvectors, orthogonality and diagonalization. (Computational and Algebraic Skills). 2) Use visualization, spatial reasoning, as well as geometric properties and strategies to model, solve problems, and view solutions, as well as conceptually extend these results to higher dimensions. (Geometric Skills).