Lectures' log (geometry), a.y. 2024-2025
Lecture |
Date |
Contents |
Reference to notes and more |
| 1 | 2/25 | 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. |
1.1, 1.2 2.3 |
| 2-3 | 2/26 | 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. |
2.1,
2.2 3.1 |
| 3/3 | Exercise
session with dr. Gollinucci: exercises
and slides
|
||
| 4 | 3/4 | 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. |
3.1, 3.2 4.1, 4.3 |
| 5-6 | 3/5 | 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. |
4.3 5.1, 5.2, 5.3, 5.4 6.1 (beginning) |
| 7 | 3/11 | 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. |
6.1,
6.2 |
| 8-9 | 3/12 |
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). |
6.3 7.1, 7.3, 7.4 |
| 3/17 | Exercise
session with dr. Gollinucci: exercises
and slides
|
||
| 10 | 3/18 | 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. |
7.5 8.1 |
| 11-12 |
3/19 | 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. |
8.1,
8.2, 8.3 |
| 13 |
3/25 | Parametric
equations of lines and planes in space, examples. Relative position of two lines in space, examples. |
9.1,
9.2 |
| 14-15 |
3/26 | 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. |
10.1,
10.2, 10.3 |
| 3/31 | Exercise
session with dr. Gollinucci: exercises
and slides
|
||
| 16 | 4/1 |
Exercises
from worksheet 5. Distance of two sets, definition. Distance of a point from a plane, examples. |
11.1,
part of 11.2 |
| 17-18 | 4/2 |
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) |
11.2,
11.3, 11.4 |
| 19-20 | 4/5 | 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.). |
12.1,
12.2,12.3 part of 13.1 |
| 21-22 | 4/9 | 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. |
13.1,
13.2 14.1, part of 14.2 |
| 4/12 |
1st midterm | ||
| 4/14 | Exercise
session with dr. Gollinucci: exercises
and slides
|
||
| 23-24 | 4/16 |
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. |
14.2 15.1, 15.2, 15.3, 15.4 |
| Easter
& Liberation day break 4/18-->4/25 |
|||
| 25 | 4/29 |
Linear
maps: definition, examples, some properties. Definition of
Kernel and Image. |
16.1,
16.2 |
| 26-27 | 4/30 |
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. |
16.2,
16.3, 16.4 17.1 |
| 5/5 | Exercise
session with dr. Gollinucci: exercises
and slides
|
||
| 28 | 5/6 |
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. |
17.1,
17.2 |
| 29-30 | 5/7 |
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. |
17.2,
17.3 18.1 |
| 31 |
5/13 |
Eigenvalues,
eigenvectors, eigenspaces, characteristic polynomial:
definition, properties, how to find them, examples. |
18.1,
18.2 |
| 32-33 |
5/14 |
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. |
19.1,
19.2, 19.3 |
| 5/19 | Exercise
session with dr. Gollinucci: exercises
and slides
|
||
| 34 |
5/20 |
Inner
products: definition and examples. Cauchy-Schwartz
inequality. Orthogonal and orthonormal sets and bases.
Gram-Schmidt orthonormalization algorithm, introduction. |
20.1,
20.2 |
| 35-36 |
5/21 |
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 ) |
20.2,
20.3, 20.4 |
| 37 |
5/27 |
Orthogonal
matrices and rotations. Quadratic forms and symmetric matrices, positive/negative definite/semidefinite, indefinite forms. |
20.3 21.1, part of 21.2 |
| 38-39 |
5/28 |
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. |
21.2 22.1, 22.1 23.1 |
| Exercise
session with dr. Gollinucci: exercises
and slides
|
|||
| 5/31 |
2nd
midterm |
||
| Republic
day 6/2 |
|||
| 40 |
6/3 |
Classification
of degenerate and non-degenerate conics through their
associated matrices, examples. Introduction to quadric surfaces. Spheres: introduction and definition. |
23.2,
23.2 24.1 |
| 41 |
6/4 |
Spheres
and circles in space: definitions and examples. Tangent
plane and tangent lines to a sphere, intersection of two
spheres, radical plane, examples. |
24.1,
24.2, 24.3 |
| Exercise
session with dr. Gollinucci: exam simulation 3, slides |
|||
| 42 | 6/6 | Exam simulation 4 | |