Eigenfunctions and eigenvalues of an LTI system.
Study of ordinary differential equations, including modeling of physical problems and interpretation of their solutions. Standard solution methods for single first-order equations, including graphical and numerical methods. Higher-order forced linear equations with constant coefficients. Complex numbers and exponentials. Matrix methods for first-order linear systems with constant coefficients. Non-linear autonomous systems; phase plane analysis. Fourier series; Laplace transforms.
This module provides a brief overview and review of the importance of eigenvectors and eigenvalues in analyzing and understanding LTI systems.
Este modulo nos da un pequeño repaso de la importancia de los eigenvectores y eigenvalores en el análisis y entedimiento de los sistemas LTI.
Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212)
The class will cover mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from 3.012 to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, fourier analysis and random walks.
