Lectures' log (geometry), a.y. 2023-2024
Lecture |
Date |
Contents |
Reference to notes and more |
| 1 | 3/5 | 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 | 3/6 | 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 |
| 4 | 3/8 | 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. |
3.1,
3.2 4.1 |
| 3/11 | Exercise session led by Dr. Canino | ||
| 5 | 3/12 | 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. |
4.2 5.1 Homework solution |
| 6-7 | 3/13 | Matrix
equations and solutions: computation of inverse matrix via
row reduction. Invertible matrices have maximal rank. Row
and column rank, rank of the transpose. Examples. Determinants: submatrices, cofactors, examples. Laplace expansion along rows and columns. Determinant of the transpose matrix. Binet's theorem. Invertible matrices have nonzero determinant. Adjugate and cofactor matrix, computation of the inverse matrix using the adjugate, examples. |
5.2,
5.3 6.1, 6.2, 6.3 |
| 8 | 3/19 | 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 the parallelogram rule. Quizzes on matrices and linear systems (quizzes file 1 without solutions). |
71.
7.3, half of 7.4 |
| 9-10 | 3/20 | Sum
and difference of vectors, multiplication by a scalar,
properties and geometric interpretation. 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. |
7.4,
7.5 8.1 |
| 3/25 | Exercise session led by Dr. Canino | ||
| 11-12 |
3/27 | 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. Parametric equations of lines and planes in space, examples. |
8.2,
8.3 9.1, 9.2 |
| 3/28-4/3:
Easter break |
|||
| 13 |
4/9 | Relative
position of two lines in space. Cartesian equations of
planes in space; switching from parametric to Cartesian
equation and backwards. Examples and exercises. |
9.1 10.1 |
| 14-15 |
4/10 | 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. |
10.2,
10.3 Homework solution |
| 4/15 | Exercise session led by Dr. Canino | ||
| 16 | 4/16 |
Distance between two sets, definition. Distance of a point
from a plane and from a line, examples. |
11.1,
11.2 |
| 17-18 | 4/17 |
Distance of a plane from another plane and from a line,
examples. Distance between two skew lines, examples. Introduction to vector spaces. Quizzes on lines and planes and distances (quizzes file 3 without solutions). |
11.3,
11.4 Introduction to 12.1 |
| 19-20 | 4/24 | Vector
spaces and subspaces: definition, properties, examples.
Union and intersection of subspaces, sum of subspaces,
examples. Linear combinations, generators, finitely
generated vector spaces: definitions and examples. |
12.1,
12.2, 12.3 13.1 |
| 21 | 4/26 | Linear
combinations, generators, span of a set of vectors: more
examples. Linearly dependent and independent vectors,
examples. Introduction to the discarding algorithm. |
13.1,13.2 Part of 14.1 |
| 4/27:
Midterm #1 |
|||
| 4/29 | Exercise session led by Dr. Canino | ||
| 22 | 4/30 | The discarding algorithm, examples. Bases and components with respect to a basis, examples. | 14.1, 14.2 |
| 5/1:
May Day |
|||
| 23 | 5/7 |
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 subspaces, examples. |
14.2 15.1, 15.2 Homework solution |
| 24-25 | 5/8 |
Examples
of computation of dimension of subspaces. Grassmann formula,
direct sum. Dimension and matrix rank: row and column space
of a matrix. Linear maps: definition, examples, some
properties. Definition of Kernel and Image. |
15.3,
15.4 16.1 |
| 5/13 | Exercise session led by Dr. Canino | ||
| 26 | 5/14 |
Some
more examples of linear maps. Linear maps K^n-->K^m and
their associated matrices. Quizzes on vector spaces (file 4 without solutions) . |
16.1 |
| 27-28 | 5/15 |
More
on 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 |
| 29 |
5/21 |
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.
Introduction to matrix of change of basis. |
17.2 |
| 30-31 |
5/22 |
Matrix
associated to the composition of linear maps and to the
inverse. Endomorphisms. Matrix of change of basis, examples.
Eigenvalues, eigenvectors, eigenspaces, characteristic
polynomial: definition, properties, how to find them,
examples. Informal notes on change of basis in a vector space (you can only read sections 1 2 and 3 so far!) |
17.3 18.1, 18.2 |
| 32-33 |
5/29 |
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 |
| 34 |
5/31 |
Inner products (dot products), definition and examples. Quizzes on linear maps, associated matrices, diagonalization (file 5 without solutions). |
20.1 |
| 6/1:
Midterm #2 |
|||
| 6/3 | Exercise session led by Dr. Canino | ||
| 35 |
6/4 |
Inner
products: Cauchy-Schwartz and triangle inequality.
Orthogonal and orthonormal sets and bases. Gram-Schmidt
orthonormalization algorithm, examples. Special and
non-special orthogonal matrices. |
20.1,
20.2, 20.3 |
| 36-37 |
6/5 |
Orthonormal
diagonalization for symmetric matrices. Quadratic forms and symmetric matrices, positive/negative definite/semidefinite, indefinite forms. Descartes' rule of signs. |
20.4 21.1, 21.2 |
| 38 |
6/7 |
More
on quadratic forms and their character of definition, with
examples. Conics in their canonical form: hyperbola, ellipse, parabola. |
21.2
22.1 |
| 6/10 | Exercise session led by Dr. Canino | ||
| 39 |
6/11 |
Conics
and rototranslations in the Euclidean plane. Reduction of a
conic in its canonical form. Classification of degenerate
and non-degenerate conics. |
22.2 23.1, 23.2 |
| 40-41 |
6/12 |
Classification of degenerate and non-degenerate conics,
examples. Introduction to quadric surfaces. Quizzes on quadratic forms, conics, orthogonal diagonalization (file 6 without solutions) Solutions of exam simulation #4 |
23.2 |