# Lectures 2019

4-3-19: Linear equations and linear systems. Solutions. Consistency of a system.
7-3-19: Basic and free variables. Matrix of coefficients. Augmented matrix. Row reduction to echelon matrix.
8-3-19: Exercises on linear systems.
11-3-19: Numerical vectors. Addition and multiplication by scalars. Linear combinations. Linear systems and vectors.
15-3-19: Linearly independent vectors. Finding subsets of linearly independent vectors.
16-3-19: Linear systems in matrix form. Exercises on linear systems in vector form.
18-3-19: Canonical basis. Linear space. Basis and coordinates of vectors.
21-3-19: Steinitz lemma. Dimension of linear spaces. Rank of a matrix.
22-3-19: Linear spaces of rows and columns of a matrix. Null space of a matrix.
25-3-19: Matrix transformations. Injectivity, surjectivity and rank.
28-3-19: Linear transformations and matrices. Examples.
29-3-19: Multiplication and addition of matrices and their linear transformations.
1-4-19: Invertible matrices.
4-4-19: Computing the inverse via row reduction
5-4-19: Change of coordinates and matrices
8-4-19: Vector (linear) spaces. Examples of polynomials and matrices.
11-4-19: Linear subspaces. Intersection of linear subspaces.
12-4-19: Exercises
15-4-19: Sum of linear subspaces. Grassmann formula.
18-4-19: Basis for intersections and sums of linear spaces.
29-4-19: Midterm exam
2-5-19: Determinants: definition, properties, computation.
3-5-19: Computation of the rank using determinants. Computation of the inverse matrix using determinants. Determinant of a product. Cramer's formula.
6-5-19: Linear transformation between vector spaces. Image and kernel.
9-5-19: Matrix of a linear transformation with respect to basis of the domain and of the range.
10-5-19: Lines in the plane and in 3-dim. space. Planes in the 3-dim. space. Cartesian and parametric equations. Lines through 2 points. Plane through 3 non collinear points. Relative position of two planes.
13-5-19: Relative position of two lines in 3-dimensional space.
16-5-19: Inner product. Norm. Distances. Orthogonal vectors, lines, planes. Angles.
17-5-19: Cross product in 3-dim. space. Mixed product. Area of parallelogram. Volume of parallelepiped.
20-5-19: Eigenvalues and eigenvectors. Characteristic polynomial.
23-5-19: Algebraic and geometric multiplicities
24-5-19: Diagonalization of endomorphisms and matrices
30-5-19: Orthogonal subspaces, orthonormal basis, orthogonal matrices.
31-5-19: Gram-Schmidt orthonormalization. Formula for the orthogonal projection.
3-6-19: Matrix of orthogonal projections. Spectral theorem for symmetric matrices.
6-6-19: Quadratic forms and their classification
7-6-19: Conic curves: classification
10-6-19: Rotations and translations that put a conic in normal form
13-6-19: Exercises
14-6-19: Exercises