Matrices with real coefficients. Opposite and transpose of a matrix, properties. Square matrices: diagonal, triangular, symmetric, skew-symmetric. Linear algebra on a number field different from R.

Matrix addition and scalar multiplication: definition and properties, examples. Product of matrices: definition and examples. Properties of the product of matrices. Invertible matrices and their properties: inverse of the transpose and the product, examples. Linear equations and linear systems. Matrices associated to a linear system and matrix form.

Exercise session with dr. Gollinucci: exercises and slides

Linear equations, linear systems, solutions, associated matrices, examples. Homogeneous and compatible systems. Elementary row operations, pivots, row-echelon form. Equivalent matrices and equivalent linear systems. Examples of Gauss reduction. Rank of a matrix.

Rank of a matrix. Examples and exercises on row reduction and rank computation. Solving a linear system: equivalent systems have equivalent associated matrices, Rouché-Capelli theorem, examples. Matrix equations and solutions: computation of inverse matrix via row reduction. Invertible matrices have maximal rank. Row and column rank, rank of the transpose. Submatrices, definition and examples.

Determinants: submatrices, cofactors, examples. Determinant of the transpose matrix. Determinant and elementary row and column operations. Laplace expansion along rows and columns. Binet's theorem. Invertible matrices have nonzero determinant. Skew-symmetric odd size matrices have zero determinant.

Adjugate and cofactor matrix, computation of the inverse matrix using the adjugate, examples. Geometric vectors: segments, applied vectors, direction, orientation and length of a vector. Parallel and coplanar vectors. Cartesian systems of coordinates. Operations on vectors: sum and difference of vectors, multiplication by a scalar, properties and geometric interpretations. Quizzes on matrices and linear systems ( quizzes file 1 without solutions ).

Exercise session with dr. Gollinucci: exercises and slides

Normalization of a vector. Characterization of parallel and coplanar vectors through rank. Dot (scalar) product: definition, examples, properties. Dot product and angles: parallel and orthogonal vectors.

Chauchy-Schwartz and triangle inequality. Orthogonal projection. Examples and exercises. Cross (vector) product: definition, geometric interpretation, properties, examples. Cross product and area of a triangle. Mixed product and volume of a tetrahedron. Some exercises on vectors. Introduction to parametric and cartesian equations of lines and planes.

Parametric equations of lines and planes in space, examples. Relative position of two lines in space, examples.

Cartesian equations of planes in space; switching from parametric to Cartesian equation and backwards. Relative position of two planes in space and Cartesian equations of lines; switching from parametric to Cartesian equation and backwards. Relative position of linear objects in the space: a line and a plane and 2 lines. Examples and exercises.

Exercise session with dr. Gollinucci: exercises and slides

Exercises from worksheet 5. Distance of two sets, definition. Distance of a point from a plane, examples.

Distance of a point from a line, distance of a plane from another plane and from a line, distance between two lines, examples. Quizzes on lines and planes and distances ( quizzes file 3 without solutions )

Vector spaces and subspaces: definition, properties, examples. Union and intersection of subspaces, sum of subspaces, examples. Linear combinations. (Online lesson, you can find the recording on the teaching portal.).

Linear combinations, generators, finitely generated vector spaces: definitions and examples. Linearly dependent and independent vectors, examples. The discarding algorithm, examples. Bases and components with respect to a basis, examples.

1st midterm

Exercise session with dr. Gollinucci: exercises and slides

Extract a basis from a set of generators and complete a set of independent vectors to a basis. Number of vectors in a basis and dimension of a vector spaces, examples. Dimension of vector subspaces, examples. Grassmann formula, direct sum. Dimension and matrix rank: row and column space of a matrix.

Easter & Liberation day break 4/18-->4/25

Linear maps: definition, examples, some properties. Definition of Kernel and Image.

Linear maps: kernel and image. Linear maps K^n-->K^m and their associated matrices, kernel, image and rank. Isomorphisms, isomorphic vector spaces, invertible matrices, examples. Linear maps for finitely generated vector spaces, examples.

Exercise session with dr. Gollinucci: exercises and slides

Linear maps for finitely generated vector spaces, matrix associated to a linear map: definition and examples. Dimension theorem: kernel, image and rank of the associated matrix.

Matrix associated to the composition of linear maps and to the inverse. Endomorphisms. Matrix of change of basis, examples. Eigenvalues, eigenvectors, eigenspaces: definition. Quizzes on vector spaces and subspaces.

Eigenvalues, eigenvectors, eigenspaces, characteristic polynomial: definition, properties, how to find them, examples.

Algebraic and geometric multiplicity of eigenvalues. Diagonalizable matrices. Similar matrices. Symmetric matrices are diagonalizable. Cayley-Hamilton theorem and eigenvalues of nilpotent matrices. Examples and exercises on diagonalization.

Exercise session with dr. Gollinucci: exercises and slides

Inner products: definition and examples. Cauchy-Schwartz inequality. Orthogonal and orthonormal sets and bases. Gram-Schmidt orthonormalization algorithm, introduction.

Orthogonal and orthonormal sets and bases. Gram-Schmidt orthonormalization algorithm, examples. Special and non-special orthogonal matrices. Orthonormal diagonalization for symmetric matrices, examples. Quizzes on linear maps, associated matrices, diagonalization ( quizzes file 5 without solutions )

Orthogonal matrices and rotations. Quadratic forms and symmetric matrices, positive/negative definite/semidefinite, indefinite forms.

More on quadratic forms and their character of definition, with examples. Descartes' rule of signs. Conics in their canonical form: hyperbola, ellipse, parabola. Conics and rototranslations in the Euclidean plane. Reduction of a conic in its canonical form.

Exercise session with dr. Gollinucci: exercises and slides

2nd midterm

Republic day 6/2

Classification of degenerate and non-degenerate conics through their associated matrices, examples. Introduction to quadric surfaces. Spheres: introduction and definition.

Spheres and circles in space: definitions and examples. Tangent plane and tangent lines to a sphere, intersection of two spheres, radical plane, examples.

Exercise session with dr. Gollinucci: exam simulation 3, slides

Exam simulation 4