Civil Engineering

GIOVANNI CERULLI IRELLI

Techniques to compute solutions of linear equations, determinants, inverse of a matrix, characteristic polynomial, eigenvalues and eigenvectors of a matrix.
Use of geometric vectors to solve problems in the geometry of the two and the three dimensional euclidean spaces.
Work with abstract vectors and vector spaces.
Knowledge of definitions and their uses in the proof of abstract theorems.

Trigonometry. Basic notion of Euclidean geometry. Basic notions on polynomials.

These arguments are recalled and discussed in the online course (in italian):

https://elearning.uniroma1.it/course/view.php?id=11798

Definition of matrix. Adjacency matrix of an oriented graph. Definition of sum of matrices. Matrix nothing. Opposing matrix. Properties of the sum of matrices. Multiplication for a scalar and its properties. Adjacency matrix of a non-oriented graph. Definition of symmetric matrix. Linear equations. Linear systems. Matrix of the coefficients of a linear system. Matrix complete with a linear system. Equivalent systems. Two matrices are said to be equivalent by rows if one is obtained from the other by elementary operations on the rows. If two systems have equivalent complete matrices then they are equivalent. Scale matrices. Small scale matrices. Gauss algorithm for scale reduction of a matrix. Use of MATLAB for the solution of linear systems. Rank of a matrix. The solutions of a mxn compatible linear system with associated matrix of rank r depend on n-r parameters. Uniqueness of the reduced-scale form of a matrix. Definition of rank of a matrix. 3 possibilities for a linear system: incompatible, with a single solution, with infinite solutions. Homogeneous systems. A homogeneous system with number of unknowns greater than the number of equations allows infinite solutions. Use of linear systems in the study of road flow networks. Use of linear systems in the study of electrical networks. Basic solutions of a homogeneous system. Each solution of a homogeneous system is a linear combination of the basic solutions. Basic solutions of a homogeneous system. Scalar product between row matrices or between column matrices. Product rows for columns of two matrices (of compatible size). Identity matrix. Product properties rows by columns. Powers of a matrix. Nilpotent matrices. Exponential of a matrix. Structure theorem for linear systems. Multiplication of block matrices. Powers of the adjacency matrix of an oriented graph. Reversible and inverse matrices of a matrix. Inverse of an invertable matrix of size 2x2. Determinant of a 2x2 matrix. If the coefficient matrix of a linear system is invertible then the system admits a single solution. Reversal algorithm. Equivalent invertibility conditions for a square matrix. A is invertible if and only it is square and there exists C such that AC = 1; if there exists C such that AC = 1 and CA = 1, then A is square. Inverse of a product; inverse of the transposed. LU decomposition and LU algorithm. Using the LU decomposition of a matrix for the solution of a linear system. Decomposition LU in case it is necessary to use a row exchange.
Elementary matrices. Inverse of a matrix as a product of elementary matrices. A matrix is ​​invertible if and only if it is produced of elementary matrices. Use of elementary matrices for the determination of the rank of a matrix. Given a matrix A there are invertible matrices T and V such that the product TAV is equal to the block matrix having as upper left block the identity matrix of size rxr, where r = rg (A). Algorithm to determine T and V matrices with this property. Exercises on LU decomposition and on the calculation of the inverse.

Vector geometry of the plane and space

Geometric vectors. Equality of geometric vectors. Matrix associated to a geometric vector (the matrix of coordinates in a given Cartesian system). Sum of geometric vectors. Parallelogram rule; tip-to-tail method. Difference of geometric vectors. Product of a geometric vector for a climb. Norm of a geometric vector. Vector interpretation of the midpoint between two points. Parallel vectors. Law of the cosines. Angle between two vectors. Orthogonal vectors. Orthogonal projection of a vattore on a non-null vector. The orthogonal projection of v on d is the parallel vector at d closest to v (analytical demonstration). The orthogonal projection of v on d is the parallel vector at d closest to v (geometric proof). Triangle area using orthogonal projection. Parametric and Cartesian equations of a straight line in the plane. Second race of linear systems. Point-to-line distance in the plane. Parametric and Cartesian equations of a straight line in space. Parametric and Cartesian equations of a plane in space. Vector product. Mixed product. Point-to-floor distance.
Formal properties of the vector product. Lagrange identity. Vector product standard as a parallelogram area. Convex sets: definition, convex combinations, the smallest convex set containing n data points is the set of convex combinations of these points (called the convex envelope). Convex envelope, affine envelope and linear envelope of n points of the plane or space. A related subset is the affine envelope of n points and a linear subspace is the linear envelope of n points. Each related subset of the plane or space is the translation of a linear subspace. Mass center of a system of n particles of variable masses in the plane or in space. Description of the center of mass in the case of equal masses. Discussion of the case of three particles of equal masses in a generic position. Triangles.

Determinants

Determinants (definition through the development of Laplace with respect to the first line). Laplace theorem: the determinant is equal to the development of Laplace with respect to any row and any column (without proof). How the determinant changes by performing an elementary operation on the rows or columns. Determinant of a triangular matrix. Det (1) = 1. A matrix is ​​invertible if and only if it has a non-zero determinant. The determinant of kA is k ^ nA, if A is a matrix nxn and k a scalar. The determinant does not change if the transposition is carried out. Binet or product theorem. Added matrix.
Addition formula and formula for the inverse of a matrix by its addition. Cramer formula for the solution of a square system with an invertible coefficient matrix. Determinant of a triangular block matrix. Solutions of some weekly exercises. Mutual position of two lines in space. Intersection between a line and a plane in space.
The determinant of a 3x3 matrix as a mixed product of the columns (or rows) of the matrix. The determinant of a 3x3 matrix as the volume of the parallelepiped generated by the columns (or rows) of the matrix.

Linear transformations of the plane

Matrix transformations of the plan. Projection and reflection with respect to a straight line (for the origin) of fixed slope. Linear transformations of the plane. A transformation of the plane is linear if and only if it is matrix. A linear transformation of the plane is uniquely determined by the values ​​it takes on the standard base vectors i and j. Matrix associated with a linear transformation of the plane. Plan rotations. Effect of a linear transformation on the standard unit square of the plane. Examples: dilatation and compression along the axis of the X; Positive and negative X-cut. Determinant of a 2x2 matrix as an area (with sign) of the parallelogram of the columns. Determinant of a 2x2 matrix as the quotient between the area of ​​the transformation of a parallelogram through the matrix and the area of ​​the parallelogram itself. Composition of linear transformations of the plane. Inverse of a linear transformation of the plane. Computer graphics: realization of the translations as multiplications for 3x3 matrices. Plan isometrics: a linear transformation of the plane is an isometry if and only if it is a rotation (around the origin) or a reflection (with respect to a straight line passing through the origin). Evaluation of the OPIS course.

Diagonalization

Motivation for diagonalization: linear dynamic systems (example). Powers of a diagonal matrix. Diagonalizable matrices (definition). Autovectors and eigenvalues. Spectrum of a matrix and real spectrum. The triangular matrices have at least one real eigenvector. The line generated by an eigenvector is stable due to the action of the matrix. The rotations of the plane (other than plus or minus 1) do not have real eigenvectors. Characteristic polynomial of a matrix (definition). Characteristic polynomial of a 2x2 matrix. Characteristic polynomial of a 3x3 matrix in terms of the trace. The characteristic polynomial of a matrix nxn is a polynomial of degree n, having a coefficient of director 1, coefficient of degree n-1 the trace and known term equal to the determinant (without proof). The spectrum of a matrix is ​​the set of zeros of the characteristic polynomial. Eigenspaces. Fundamental theorem of algebra (without proof). Linearly independent vectors of R ^ n. An nxn matrix having n distinct eigenvalues ​​is diagonalizable on R. Algebraic and geometric multiplicity of an eigenvalue. Geometric multiplicity is less than or equal to algebraic multiplicity. Still on linear independence in R ^ n. Bases of R ^ n. Accent algorithm at a basis of R ^ n. An nxn matrix is ​​diagonalizable on R if and only if there exists a basis of R ^ n composed of eigenvectors for the matrix. A matrix nxn is diagonalizable on R if and only if the spectrum of the matrix is ​​real and for each of its eigenvalues ​​the algebraic multiplicity is equal to the geometric multiplicity.

The Vector Space R ^ n

Fundamental theorem on the dimension. Bases of vector subspaces of R ^ n. Similar matrices (two similar matrices have the same rank, the same determinant and the same characteristic polynomial). Cayley-Hamilton's theorem. Vector subspaces of R ^ n. Basis of vector subspaces. Each vector subspace of R^n admits a base. The Kernel of a matrix is a vector subspace whose base is composed of the basic solutions of the homogeneous system associated to the matrix. Orthogonal complement. The orthogonal complement of a vector subspace is a vector subspace. Each vector subspace of R ^ n is the Kernel of a matrix.

The Vector Space R^n: metric structure

Orthogonal and orthonormal sets of vectors in R ^ n. Pythagorean theorem. Fourier development (compared to an orthogonal base). Gram-Schmidt algorithm. Each vector subspace of R ^ n admits an orthonormal basis. Orthogonal projection. QR decomposition. Approximate solutions of non-solvable systems. Polynomial interpolation: Interpolating polynomial of n pairs of data. Least squares approximation: determination of the polynomial of degree m that best approximates n pairs of data, in the sense of least squares. Orthogonal matrices. Theorem of the principal axes (or spectral theorem).

Vector spaces

Singular values ​​of an mxn array. Decomposition to the singular values ​​of an mxn matrix. Rule of an m-n matrix. Rayleigh quotient of a symmetric matrix. The norm of a matrix is ​​equal to the largest singular value of the matrix. Given a mxn matrix of rank r, there exists a matrix of rank k (for each 0m they are linearly dependent). Dimension. The {cos (mx)} functions are linearly independent (in particular the function space from [0.2pi] to R does not allow a base). A set of polynomials of different degrees is linearly independent. Completion theorem at a base. A base is a maximal set of linearly independent vectors. Linear transformations. Core and image of a linear transformation. A linear transformation is injective if and only if it has a trivial nucleus. Theorem of dimensions. A linear transformation between spaces of the same size is injective if and only if it is injective. A vector space of size n is isomorphic to R ^ n. A linear transformation is an isomorphism if and only if it sends bases in bases. A linear transformation is an isomorphism if and only if it admits an inverse. Matrix associated with a linear transformation, with respect to the choice of two bases (one departing and one arriving). If the linear transformation is the identity, this matrix is ​​called the base change matrix. Definition of product scalar in a vector space. A vector space with a scalar product is called a metric space. Fourier coefficients. Angle between vectors of a metric space. Internal scalar product (induced by a base). The scalar products of R^n are induced by positive definite symmetric matrices.

W. Keith Nicholson: "Linear algebra with applications". This is free and can be downloaded from the website

https://lyryx.com/linear-algebra-applications/

I recommend to subscribe to Lyryx to get the full package. It is worth its price.

W. Keith Nicholson: "Linear algebra with applications". This is free and can be downloaded from the website

https://lyryx.com/linear-algebra-applications/

I recommend to subscribe to Lyryx to get the full package. It is worth its price.

Lectures on blackboard. Exercise sessions. Office hours.

Four two hours long lectures per week. Two hours per week of exercise sections. Two hours per week of office hours.

Written and oral exam

See the webpage

http://www.sbai.uniroma1.it/~giovanni.cerulliirelli/didattica/

• Linear systems


• Affine geometry of the euclidean plane


• Vector spaces


• Linear maps


• Determinant


• Diagonalization


• Orthogonal diagonalization and spectral theorem


• Bilinear forms and Sylvester theorem


• Euclidean geometry of the plane and the space


• Euclidean space