Linear Algebra and Geometry
Timetable of the lectures: Mon 4pm, Thu 11:30am, Fr 9:30am
Microsoft Teams, team "Linear Algebra and Geometry". Code of the Team:
1_February 28: Linear equations and linear systems. Solutions. Consistency of a system.
2_March 3: Basic and free variables. Matrix of coefficients. Augmented matrix. Row reduction to echelon matrix.
3_March 4: Reduced echelon form of a matrix. Linear systems with parameters
4_March 7: exercises on linear systems. Numerical vectors. Addition and multiplication by scalars. Linear combinations.
5_March 10: Vector Spaces.
Implicit and parametric equations of lines and planes.
6_March 11: Vector subspaces and generators. Linear span.
7_March 14: A linear system is consistent if and only if the constant terms vector is a linear combination of the columns
of the coefficients matrix. Linear independence. Minimal list of generators.
March 17: tutorial (Prof. Pareschi)
8_March 18: maximal linearly independent subset. Finding subsets of linearly
9_March 21: uniqueness of the coefficients in a linear combination of
linearly independent vectors.
Multiplication and addition of matrices. Vector spaces of matrices.
10_March 24: The solutions of a consistent linear system in n variables are
the translate of a vector subspace of R^n. Finite dimensional vector spaces.
In a finite dimensional vector space, the length of a linearly independent
spanning set in lesser or equal to the
length of any spanning set.
In a finite dimensional vector space, any two linearly independent spanning
set have the same length.
March 25: tutorial
11_March 28: Basis and dimension of a finite dimensional vector space.
Canonical basis and dimension of R^n.
Any finite dimensional vector space has (at least) a basis.
Any spanning list contains a basis.
Any linearly independent list can be extended to a basis.
12_March 31: The vector space of real polynomials in one variable.
A vector space with finite dimension n contains subspaces of any
dimension between 0 and n.
Moreover, any proper subspace has dimension
13_April 1: Transpose of a matrix. Linear subspaces spanned by the rows
(or the columns) of a matrix.
In a matrix M = AB, any column is in col(A), any row is in row(B). Rank of
Theorem on the solutions of a linear system
14_April 4 (Prof. Santi): Linear transformations, their relationship with
matrices. Image and Kernel of a linear transformation, their
relationship with injectivity and surjectivity. Examples and exercises.
15_April 7 (Prof. Santi): Exercises on linear transformations, Image and
Kernel. Resumè on subspaces, basis and coordinates.
Dimension Theorem, its consequences and exercises.
Intersection and sum of two vector subspaces, and their basis. Exercises
17_April 11 (Prof. Santi): Product of matrices, its main properties
and its relationship with the composition of linear transformations.
Main properties of transposition of matrices.
Invertible linear transformations and invertible matrices,
product and transposition of invertible matrices.
Examples, including the shear transformations in the plane.
18_April 14 (Prof. Santi): Computing the inverse matrix via row reduction.
Inverse and determinant of a 2 by 2 matrix.
Linear systems with invertible matrix of coefficients.
Exercises on linear transformations.
19_April 15: relation between the dimension of two vector subspaces, the
intersection and the sum. Exercises
20_April 21: Midterm exam in presence
21_April 22: Oriented Lenghts and vector spaces in geometry.
Frame in the Euclidean Space. Coordinates.
22_April 28: (Prof. Santi) Determinant: defining properties and Laplace
Minors and cofactors. Determinant and Gauss reduction. Examples (determinant
of diagonal and upper triangular matrices) and exercises.
23_April 29: Standard scalar product in R^n. Orthonormal basis.
24_May 2: (Prof. Santi) Determinant of a product: Binet Theorem.
Computation of the inverse matrix using determinants. Cramer's formula.
Geometric applications: area of a parallelogram and volume of a parallelepiped
using determinants. Examples and exercises.
May 3: Tutorial (Prof. Sabatino)
25_May 5 (Prof. Santi): Computation of the rank using determinants. Examples.
Application to resolutions of general linear systems,
using square submatrices and Cramer's formula.
26_May 6: Lines in the plane and in 3-D space.
Planes in the 3-D space. Cartesian and parametric equations.
Lines through 2 points.
Parallelism and orthogonality with a line or a plane in the
Euclidean 3D space. Methods to find an orthonormal basis with the first
element parallel to a given direction. Orthogonal projection and rejection
of a non zero vector along another non zero vector.
How to determine a non zero solution of a homogeneous linear system
of rank 2 in 3 variables.
27_May 9 (Prof. Santi): Change of bases and coordinates: basic definitions,
matrix of basis change and its properties. Examples.
May 10 (Prof. Sabatino): Tutorial
28_May 12(Prof. Santi): Linear transformations between vector spaces, their
kernel and image. Matrix of a linear transformation with respect to
given bases of the domain and the codomain. Change of coordinates
and matrices. Similar matrices. Examples and exercises.
29_May 13 (Prof. Santi): The field of complex numbers:
basic definitions, Gauss plane, de Moivre formula, n-th roots of
a complex number, fundamental Theorem of Algebra. Examples and exercises.
30_May 16 (Prof. Santi): Vector subspaces invariant
for an endomorphism of a vector space.
Eigenvectors and eigenvalues: definitions and examples.
Theorem: eigenvectors relative to different eigenvalues
are linearly independent
(with an idea of the proof). Characteristic polynomial
of an endomorphism of a vector space. Examples.
May 17 (Prof. Sabatino): Tutorial
31_May 19(Prof. Santi): Diagonalizable linear transformations and
diagonalizable matrices. Eigenspaces. Examples and exercises
32_May 20: Gram-Schmidt orthonormalization. Normal vector to a plane in a
3-D space. Plane through 3 non collinear points.
Cross product in 3-D space. Distance amomg a point and a plane
33_May 23(Prof. Santi): Spectrum of an endomorphism of a vector space.
and geometric multiplicities, their relationship. Exercises.
34_May 26 (Prof. Santi): Spectral theorem for real
symmetric matrices (including first half
of the proof). Eigenvectors of a real symmetric
matrix relative to different eigenvalues
are orthogonal. Examples and exercises.
35_May 27: Distance among euclidean subspaces. lines in the 3D space.
Mixed product. Area of parallelogram. Volume of parallelepiped.
Conic curves: classification
36_May 30 (Prof. Santi):
37_June 6 Conic curves: classification
38_June 9: Examples and exercises.
39_June 10: Midterm Examination in presence