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[[File:Logos-tadeo-central.png]]
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== English version ==
 
This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in models and simulation for several disciplines. It includes systems of equations, vector spaces, determinants and eigenvalues.
 
This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in models and simulation for several disciplines. It includes systems of equations, vector spaces, determinants and eigenvalues.
  
Line 8: Line 10:
 
=== Content ===
 
=== Content ===
  
−
# Linear equation systems
+
<ul>
−
## Matrix: Operations and properties.
+
<li>Vectorial calculus. One variable calculus.</li>
−
## Resolve linear systems by Gauss and Gauss-Jordan methods.
+
<ol>
−
## Resolve linear systems by inverse and Cramer's rule.
+
<li>Review of one variable calculus.</li>
−
## Applications: Leonlief product supplies analysis.
+
<li>Function definition.</li>
 +
<li>Derivatives.</li>
 +
<li>Integrals.</li>
 +
</ol>
  
−
# Rn Vectors
+
<li>First order differential equations.</li>
−
## Rn Vectors.
+
<ol>
−
## Operations between vectors.
+
<li>Basic Concepts. Modeling.
−
## Dot product, norm and projection
+
</li><li>Geometric Meaning of y r ϭ f (x, y). Direction Fields, Euler’s Method.
−
## Cross product.
+
</li><li>Separable ODEs. Modeling.
−
## Applications: Parallels and orthogonal vectors, area and surface vectors
+
</li><li>Exact ODEs. Integrating Factors.
 +
</li><li>Linear ODEs. Bernoulli Equation. Population Dynamics.
 +
</li><li>Orthogonal Trajectories.
 +
</li><li>Existence and Uniqueness of Solutions for Initial Value Problem.
 +
</ol>
  
−
# Vector space
+
<li>Second order differential equations.</li>
−
## Spaces and subspaces.
+
<ol>
−
## Lineal combination and space generation.
+
</li><li>Homogeneous Linear ODEs of Second Order
−
## lineal dependence and lineal independece.
+
</li><li>Homogeneous Linear ODEs with Constant Coefficients
−
## Basis and dimension
+
</li><li>Modeling of Free Oscillations of a Mass–Spring System
−
## Linear transformations.
+
</li><li>Differential Operators.
−
## Eigen-values and eigen-vectors
+
</li><li>Euler–Cauchy Equations
 +
</li><li>Existence and Uniqueness of Solutions.  
 +
</li><li>Nonhomogeneous ODEs
 +
</li><li>Modeling: Forced Oscillations.  
 +
</ol>
  
−
# One variable functions
+
<li>Linear Algebra and systems of ordinary differential equations</li>
−
## One variable functions generalities.
+
<ol>
−
## Several variable functions generalities.
+
<li>Spaces and subspaces.</li>
−
## Partial derivates, gradient and directional derivates.
+
<li>Lineal combination and space generation.</li>
−
## Optimization.
+
<li>lineal dependence and lineal independece.</li>
−
## Multiple integration.
+
<li>Basis and dimension</li>
−
## Vector analysis.
+
<li>Linear transformations.</li>
 +
<li>Eigen-values and eigen-vectors</li>
 +
</ol>
  
−
# Differential equations
+
<ol>
−
## First order differential equations.
+
<li>Matrix: Operations and properties.</li>
−
## Second order differential equations.
+
<li>Resolve linear systems by Gauss and Gauss-Jordan methods.</li>
−
## Differential equations systems.
+
<li>Resolve linear systems by inverse and Cramer's rule.</li>
−
## Solution by transforms methods
+
<li>Applications: Leonlief product supplies analysis.</li>
−
## Introduction to partial differential equations.
+
</ol>
  
 +
<ol>
 +
<li>systems of ordinary differential equations.</li>
 +
<li> Homogeneous systems of ordinary differential equations.</li>
 +
<li>Resolve linear systems of differential equations .</li>
 +
<li>linearization of high order differential equations.</li>
 +
</ol>
 +
 +
<li>Vectorial calculus. Several variable functions.</li>
 +
<ol>
 +
</li><li>Vectors in 2-Space and 3-Space.
 +
</li><li>Inner Product (Dot Product).
 +
</li><li>Vector Product (Cross Product).
 +
</li><li>Vector and Scalar Functions and their Fields.
 +
</li><li>Vector Calculus: Derivatives.
 +
</li><li>Functions of Several Variables.
 +
</li><li> Gradient of a Scalar Field.
 +
</li><li>Directional Derivatives.
 +
</li><li>Divergence of a Vector Field.
 +
</li><li>Curl of a Vector Field.
 +
</li><li>Lagrange multipliers.
 +
</ol>
 +
<li>partial differential equations.</li>
 +
<ol>
 +
</li><li>Introduction to partial differential equations (PDE)
 +
</li><li>Solution by transforms methods</li>
 +
</ol>
 +
</ul>
  
 
=== Week activities ===
 
=== Week activities ===
Line 49: Line 91:
 
<tr><td>Session date</td><td>Teacher</td><td>Topic</td><td>External resources</td></tr>
 
<tr><td>Session date</td><td>Teacher</td><td>Topic</td><td>External resources</td></tr>
 
<tr>
 
<tr>
−
<td>1 - 30-Ene</td><td>Todos, Darwin</td>
+
<td>1 - 31-Julio</td><td>Todos, Camilo</td>
 
<td>
 
<td>
 +
 
<ul>
 
<ul>
 
<li>Presentation</li>
 
<li>Presentation</li>
−
<li>Matrices, Vectors: Addition and Scalar Multiplication</li>
+
<li>Vectorial calculus. One variable calculus.</li>
−
<li>Matrix Multiplication</li>
+
<ol>
−
<li>Linear Systems of Equations. Gauss Elimination</li>
+
<li>Review of one variable calculus.</li>
−
<li>Solutions of Linear Systems: Existence, Uniqueness</li>
+
<li>Function definition.</li>
−
<li>Determinants. Cramer’s Rule</li>
+
<li>Derivatives.</li>
−
<li>Inverse of a Matrix. Gauss–Jordan Elimination </li>
+
<li>Integrals.</li>
 +
</ol>
 
</ul>
 
</ul>
 
</td>
 
</td>
−
<td>[http://ocw.mit.edu/ans7870/18/18.06/tools/Applets_sound/uropmovie.html Matrix Multiplicaction]<br>
+
<td>
−
[http://ocw.mit.edu/ans7870/18/18.06/javademo/GaussElim/ Gauss Elimination]<br>
+
<!--[http://www.mathresource.iitb.ac.in/linear%20algebra/appletsla.html Applets algebra lineal]<br/>
−
[http://ocw.mit.edu/ans7870/18/18.06/javademo/Determinant/ Determinants]</td>
+
[http://ocw.mit.edu/ans7870/18/18.06/tools/Applets_sound/uropmovie.html Matrix Multiplicaction]<br/>
 +
[http://ocw.mit.edu/ans7870/18/18.06/javademo/GaussElim/ Gauss Elimination]<br/>
 +
[http://ocw.mit.edu/ans7870/18/18.06/javademo/Determinant/ Determinants]<br/>
 +
[http://people.hofstra.edu/Stefan_Waner/tutorialsf1/scriptpivot2.html Gauss-Jordan Pivot trainning]<br/>
 +
[http://personal.bgsu.edu/~meel/Tools/ Algebra Lineal tutoriales]-->
 +
</td>
 
</tr>
 
</tr>
  
 
<tr>
 
<tr>
−
<td>2 6-Feb</td><td>Darwin</td>
+
<td>2 14 Agosto</td><td>Camilo</td>
 
<td>
 
<td>
 
<ul>
 
<ul>
−
<li>Linear Independence. Rank of a Matrix. Vector Space</li>
+
<li>LAB: One variable Calculus with MatLab and applications [http://www.tutorialspoint.com/matlab/matlab_plotting.htm MatLab-Plotting] [http://www.tutorialspoint.com/matlab/matlab_calculus.htm MatLab-Calculus]
−
<li>Vector Spaces, Inner Product Spaces, Linear Transformations</li>
+
[[file:Taller-CalcVec1D.pdf]]</li>
−
<li>The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors</li>
+
 
</ul>
 
</ul>
 
</td>
 
</td>
−
<td>[http://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_1.html Eigen-vectors/values]<br>
+
<td>
−
[http://ocw.mit.edu/ans7870/18/18.06/javademo/Eigen/ eigenvalues]</td>
+
<!--[http://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Linear_Independence Linear Independence Examples]<br>
 +
[http://algebra.nipissingu.ca/tutorials/vector_space.html Vector Space]<br/>
 +
[http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=li Linear independence]<br/>
 +
[http://ocw.mit.edu/ans7870/18/18.06/tools/individual/eigen_lecture_1.html Eigen-vectors/values]<br>
 +
[http://ocw.mit.edu/ans7870/18/18.06/javademo/Eigen/ eigenvalues]</td>-->
 
</tr>
 
</tr>
−
 
 
<tr>
 
<tr>
−
<td>3 13-Feb</td><td>Darwin</td>
+
<td>3 21-Agosto</td><td>Angelica</td>
 
<td>
 
<td>
 
<ul>
 
<ul>
−
<li>LAB:  Linear algebra</li>
+
<li>First order differential equations.</li>
 +
<ol>
 +
<li>Basic Concepts. Modeling.
 +
</li><li>Geometric Meaning of y r ϭ f (x, y). Direction Fields, Euler’s Method.
 +
</li><li>Separable ODEs. Modeling.
 +
</li><li>Exact ODEs. Integrating Factors.
 +
</li><li>Linear ODEs. Bernoulli Equation. Population Dynamics.
 +
</li><li>Orthogonal Trajectories.
 +
</li><li>Existence and Uniqueness of Solutions for Initial Value Problem.
 +
</ol>
 
</ul>
 
</ul>
 
</td>
 
</td>
−
<td></td>
+
<td>
 +
<!--MuPAD [http://math.uprag.edu/MuPAD.pdf 1] [http://math.uprag.edu/mupad-algebra-lineal.pdf 2] [http://www.brown.edu/Departments/Engineering/Courses/En4/Tutorials/Mupad_tutorial.pdf 3]<br/>
 +
[[File:LaboratorioUno-AlgebraLineal.pdf]] [[Taller1|Punto 3 Taller 1]]-->
 +
</td>
 
</tr>
 
</tr>
  
 
<tr>
 
<tr>
−
<td>4 20-Feb</td><td>Darwin</td>
+
<td>4 28-Agosto</td><td>Angélica</td>
 
<td>
 
<td>
 
<ul>
 
<ul>
−
<li>Test</li>
+
</li><li>Homogeneous Linear ODEs of Second Order
 +
</li><li>Homogeneous Linear ODEs with Constant Coefficients
 +
</li><li>Modeling of Free Oscillations of a Mass–Spring System
 +
</li><li>Differential Operators.
 +
</li><li>Euler–Cauchy Equations
 +
</li><li>Existence and Uniqueness of Solutions.
 +
</li><li>Nonhomogeneous ODEs
 +
</li><li>Modeling: Forced Oscillations.
 
</ul>
 
</ul>
 
</td>
 
</td>
Line 100: Line 170:
 
 
 
<tr>
 
<tr>
−
<td>5 27-Feb</td><td>Camilo</td>
+
<td>5 4-Sept</td><td>Jorge</td>
 
<td>
 
<td>
 
<ul>
 
<ul>
−
<li>Vectors in 2-Space and 3-Space</li>
+
<li>Linear Algebra and systems of ordinary differential equations</li>
−
<li>Inner Product (Dot Product), Vector Product (Cross Product)</li>
+
<ol>
−
<li>Review of one variable Calculus</li>
+
<li>Spaces and subspaces.</li>
−
<li>Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives</li>
+
<li>Lineal combination and space generation.</li>
 +
<li>lineal dependence and lineal independece.</li>
 +
<li>Basis and dimension</li>
 +
<li>Linear transformations.</li>
 +
<li>Eigen-values and eigen-vectors</li>
 +
</ol>
 
</ul>
 
</ul>
 
</td>
 
</td>
−
<td></td>
+
<td rowspan="4">
 +
<!--[http://mathinsight.org/applet/changing_surfaces_stokes_theorem Stokes theorem]<br/>
 +
[http://mathinsight.org/divergence_idea The idea of the divergence of a vector field]<br/>
 +
[http://mathinsight.org/divergence_subtleties Subtleties about divergence]<br/>
 +
[http://mathinsight.org/curl_idea The idea of the curl of a vector field]<br/>
 +
[http://mathinsight.org/curl_subtleties Subtleties about curl]<br/>
 +
[http://mathinsight.org/greens_theorem_idea The idea behind Green's theorem]-->
 +
</td>
 
</tr>
 
</tr>
  
 
<tr>
 
<tr>
−
<td>6 6-Mar</td><td>Camilo</td>
+
<td>6 11-Sept</td><td>Jorge</td>
 
<td>
 
<td>
 
<ul>
 
<ul>
−
<li>Functions of Several Variables</li>
+
<ol>
−
<li>Gradient of a Scalar Field. Directional Derivative</li>
+
<li>Matrix: Operations and properties.</li>
−
<li>Divergence of a Vector Field</li>
+
<li>Resolve linear systems by Gauss and Gauss-Jordan methods.</li>
−
<li>Curl of a Vector Field</li>
+
<li>Resolve linear systems by inverse and Cramer's rule.</li>
−
<li>Lagrange multipliers</li>
+
<li>Applications: Leonlief product supplies analysis.</li>
 +
</ol>
 +
</ul>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td>7 18-Sept</td><td>Angélica</td>
 +
<td>
 +
<ul>
 +
<li>systems of ordinary differential equations.</li>
 +
<li> Homogeneous systems of ordinary differential equations.</li>
 +
<li>Resolve linear systems of differential equations .</li>
 +
<li>linearization of high order differential equations.</li>
 
</ul>
 
</ul>
 
</td>
 
</td>
−
<td></td>
 
 
</tr>
 
</tr>
  
 
<tr>
 
<tr>
−
<td>7 13-Mar</td><td>Camilo</td>
+
<td>8 25 -Sept</td><td>Jorge and Angélica</td>
 
<td>
 
<td>
 
<ul>
 
<ul>
−
<li>Line Integrals, Path Independence</li>
+
<li>LAB: EDOs with MatLab, practice session.</li>
−
<li>Calculus Review: Double Integrals.</li>
+
 
</ul>
 
</ul>
 
</td>
 
</td>
−
<td></td>
 
 
</tr>
 
</tr>
  
 
<tr>
 
<tr>
−
<td>8 20-Mar</td><td>Camilo</td>
+
<td>9 2 -Oct</td><td>Todos</td>
 
<td>
 
<td>
 
<ul>
 
<ul>
−
<li>Green’s Theorem in the Plane</li>
+
<li>TEST</li>
−
<li>Surfaces for Surface Integrals</li>
+
−
<li>Surface Integrals</li>
+
−
<li>Triple Integrals. Divergence Theorem of Gauss</li>
+
 
</ul>
 
</ul>
 
</td>
 
</td>
Line 151: Line 239:
  
 
<tr>
 
<tr>
−
<td>9 27-Mar</td><td>Camilo</td>
+
<td>10 16 -Oct</td><td>Camilo</td>
 
<td>
 
<td>
 
<ul>
 
<ul>
−
<li>LAB: Vector calculus</li>
+
<li>Vectors in 2-Space and 3-Space.[[file:Vectors-ScalarFields.pdf]]</li>
 +
</ul>
 +
<ol>
 +
<li>Inner Product (Dot Product).</li>
 +
<li>Vector Product (Cross Product).</li>
 +
</ol>
 +
<ul>
 +
<li>Vector and Scalar Functions and their Fields.</li>
 +
</ul>
 +
<ol>
 +
<li>Vector Calculus: Derivatives.</li>
 +
<li>Functions of Several Variables.</li>
 +
</ol>
 +
<td></td>
 +
</tr>
 +
 
 +
<tr>
 +
<td>11 23 -Oct</td><td>Camilo</td>
 +
<td>
 +
<ul>
 +
<li>Gradient of a Scalar Field. Application in ecology:[[file:BIOS-83-97.pdf]] Taller: [[file:Taller-CalcVec2.pdf]]    Datos: [[file:data.txt]] Salida de zunzun.com: [[file:zunzunout.pdf]]</li>
 +
<li>Directional Derivatives.</li>
 +
<li>Divergence of a Vector Field.</li>
 +
<li>Curl of a Vector Field.</li>
 +
<li>Lagrange multipliers.</li>
 +
<li>Tutorial de cálculo vectorial con Matlab: [http://www2.math.umd.edu/~jmr/241/MATLABmaterials.html]  </li>
 
</ul>
 
</ul>
 
</td>
 
</td>
Line 160: Line 273:
 
</tr>
 
</tr>
  
 +
 +
 +
<!--
 
<tr>
 
<tr>
 
<td>10 3-Abr</td><td>Camilo</td>
 
<td>10 3-Abr</td><td>Camilo</td>
Line 244: Line 360:
 
<td></td>
 
<td></td>
 
</tr>
 
</tr>
−
 
+
-->
 
</table>
 
</table>
−
 
  
 
== Versión en español ==
 
== Versión en español ==
Line 308: Line 423:
 
## Introducción a las ecuaciones diferenciales parciales  
 
## Introducción a las ecuaciones diferenciales parciales  
  
−
== Metodología ==
+
=== Metodología ===
 
* La enseñanza de este curso se realizará a través de clases teóricas y prácticas en  salas de cómputo.
 
* La enseñanza de este curso se realizará a través de clases teóricas y prácticas en  salas de cómputo.
 
* En el desarrollo de las clases  prevalecerá  la conceptualización en los temas  a tratar  sobre las destrezas operativas que pueden trabajarse mediante Sistemas Algebraicos Computacionales SAC.  
 
* En el desarrollo de las clases  prevalecerá  la conceptualización en los temas  a tratar  sobre las destrezas operativas que pueden trabajarse mediante Sistemas Algebraicos Computacionales SAC.  
 
* Durante el curso se hará énfasis  en la importancia de cada tema en la  formulación de modelos, considerando los diferentes campos de aplicación, más que en la implementación de algoritmos.
 
* Durante el curso se hará énfasis  en la importancia de cada tema en la  formulación de modelos, considerando los diferentes campos de aplicación, más que en la implementación de algoritmos.
  
 +
== Bibliography/Bibliografía ==
  
 +
# Giordano, F., Fox, W., Horton, S., & Weir, M. (2009). '''A first course in mathematical modelling'' . Canada: Brooks/Cole Cengage Learning.
 +
# Grossman, S. (2008). '''Álgebra Lineal'''. México: Mc Graw Hill.
 +
# Kolman, B., & Hill, D. (2013). '''Álgebra Lineal. Fundamentos y apliaciones'''. Bogotá: Pearson.
 +
# Larson, R., & Edwards, B. H. (2010). '''Calculo 2 de Varias variables'''. China: Mc Graw Hill.
 +
# Meerschaaert, M. (2007). '''Mathematical Modeling'''. San Diego, United States of America: Elsevier Academic Press.
 +
# Zill, D. G. (2002). '''Ecuaciones diferenciales con aplicaciones de modelado'''. México: Cengage Learning.
  
−
=== Presentaciones ===
+
 
 +
== Slides / Presentaciones ==
  
 
# [[File:SemanaUno.pdf]]
 
# [[File:SemanaUno.pdf]]
 +
# [[File:SemanaDos.pdf]]
 +
# [[File:AlgLineal.pdf]]
 +
# [[File:taller-de-ecuaciones.pdf]]
 +
[https://sites.google.com/site/algoritmosyprogramacionuc/archivos/Laboratorio_Sistema.m?attredirects=0&d=1 link Laboratorio sistemas.m]
  
−
<!--
+
== Matlab's Scripts ==
−
+
=== Linear dependency ===
−
Criterios de evaluación:
+
<code>
−
Conforme a las políticas institucionales, los criterios de evaluación son discrecionales.  
+
  % PRUEBA PARA MOSTRAR INDEPENDENCIA LINEAL<br/>
−
Bibliografía básica:
+
  x = rand(1, 3)-0.5;<br/>
−
Giordano, F., Fox, W., Horton, S., & Weir, M. (2009). A first course in mathematical modeling . Canada: Brooks/Cole Cengage Learning.
+
  y = rand(1, 3)-0.5;<br/>
−
Grossman, S. (2008). Álgebra Lineal. México: Mc Graw Hill.
+
  a = rand(10, 1)-0.5;<br/>
−
Kolman, B., & Hill, D. (2013). Álgebra Lineal. Fundamentos y apliaciones. Bogotá: Pearson.
+
  b = rand(10, 1)-0.5;<br/>
−
Larson, R., & Edwards, B. H. (2010). Calculo 2 de Varias variables. China: Mc Graw Hill.
+
  z = a*x + b*y;<br/>
−
Meerschaaert, M. (2007). Mathematical Modeling. San Diego, United States of America: Elsevier Academic Press.
+
  hold off<br/>
−
Zill, D. G. (2002). Ecuaciones diferenciales con aplicaciones de modelado. México: Cengage Learning.
+
  quiver3(0,0,0,x(1),x(2),x(3), 'r');<br/>
 +
  hold on<br/>
 +
  t= zeros(10,3);<br/>
 +
  quiver3(0,0,0,y(1),y(2),y(3), 'b');<br/>
 +
  quiver3(t(:,1),t(:,2),t(:,3),z(:,1),z(:,2),z(:,3),'g');<br/>
 +
  title('Dependencia lineal x rojo, y azul, combinaciones aleatorias verde ')<br/>
 +
</code>
  
 +
=== Lineal transformations ===
 +
<code>
  
−
 
+
  %PRUEBAS CON EL DETERMINANTE EN MATRICES DE ESCALAMIENTO
−
 
+
  % construir el poligono con vertices x,y
−
 
+
  x = [2 2 1 1 2 3 2  2  5 5 4 5 6 6 5 5 2];
−
 
+
  y = [1 4 4 7 7 8 9 11 11 9 8 7 7 4 4 1 1];
−
 
+
  x = x-3.5;
−
 
+
  y = y-5.5;
−
 
+
  hold off
−
-->
+
  plot(x, y, 'b')
 +
  title 'poligono original'
 +
  axis([-7 7 -11 11])
 +
  grid
 +
  pause
 +
  p = [x; y;];
 +
  S1 = [1 0; 0 2]
 +
  det_s1 = det(S1)
 +
  S2 = [sqrt(2) 0; 0 sqrt(2)]
 +
  det_s2 = det(S2)
 +
  S3 = [0.5 0; 0 0.5]
 +
  det_s3 = det(S3)
 +
  S4 = S3*S2;
 +
  det_s4 = det(S4)
 +
  S5 = [0.5 0; 0 2]
 +
  det_s5 = det(S5)
 +
  p1 = S2*p;
 +
  hold off
 +
  plot(x, y, 'b:')
 +
  hold on
 +
  plot(p1(1, :), p1(2, :), 'r')
 +
  title 'poligono escalado al doble'
 +
  axis([-7 7 -11 11])
 +
  grid
 +
  pause
 +
  p2 = S4*p;
 +
  hold off
 +
  plot(x, y, 'b:')
 +
  hold on
 +
  plot(p1(1, :), p1(2, :), 'b:')
 +
  plot(p2(1, :), p2(2, :), 'r')
 +
  title 'polígono escalado a la cuarta parte'
 +
  axis([-7 7 -11 11])
 +
  grid
 +
  pause
 +
  % PRUEBAS DE ACUMULAR LA ROTACION CON LA MULTIPLICACION
 +
  a = pi/4;
 +
  R = [cos(a) -sin(a); sin(a) cos(a)]
 +
  M = [1 0; 0 1];
 +
  for i=1:8
 +
    M = R*M;
 +
    p1 = M*p;
 +
    hold off
 +
    plot(x, y, 'b:')
 +
    hold on
 +
    plot(p1(1, :), p1(2, :), 'r')
 +
    title 'polígono rotacion acumulada'
 +
    axis([-7 7 -11 11])
 +
    grid
 +
    pause
 +
  end
 +
  % SINGULAR VALUE DECOMPOSITION PARA UNA MATRIZ GENERADA CON ROTACION Y ESCALAMIENTO
 +
  R = [0.8 0.0 0.6; 0 1 0; -0.6 0.0 0.8]
 +
  S = [0.5 0 0; 0 2 0; 0 0 1.5]
 +
  M = S*R;
 +
  [U, Z, V] = svd(M)
 +
  % ANIMACION 3D CON UNA MATRIZ DE ROTACION APLICADA A PUNTOS EN UNA SUPERFICIE
 +
  hold off
 +
  n = 400;
 +
  x = 2*(rand(1, n)-0.5);
 +
  y = 2*(rand(1, n)-0.5);
 +
  z = x.^2 - y.^2;
 +
  p = [x; y; z];
 +
  for i=1:180,
 +
    a = i*pi/90;
 +
    rx = [1 0 0; 0 cos(a) -sin(a); 0 sin(a) cos(a)];
 +
    pp = rx*p;
 +
    plot(pp(1,:), pp(2,:), 'r.')
 +
    axis([-1 1 -1 1])
 +
    pause(0.1)
 +
  end
 +
</code>

Latest revision as of 14:23, 10 November 2014

Logos-tadeo-central.png

Contents

[edit] English version

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in models and simulation for several disciplines. It includes systems of equations, vector spaces, determinants and eigenvalues.

[edit] Goals

  • Enhance the mathematics skills and fundaments.
  • Give enough mathematic basics to understand the topics of modelling and simulation systems.


[edit] Content

  • Vectorial calculus. One variable calculus.
    1. Review of one variable calculus.
    2. Function definition.
    3. Derivatives.
    4. Integrals.
  • First order differential equations.
    1. Basic Concepts. Modeling.
    2. Geometric Meaning of y r ϭ f (x, y). Direction Fields, Euler’s Method.
    3. Separable ODEs. Modeling.
    4. Exact ODEs. Integrating Factors.
    5. Linear ODEs. Bernoulli Equation. Population Dynamics.
    6. Orthogonal Trajectories.
    7. Existence and Uniqueness of Solutions for Initial Value Problem.
  • Second order differential equations.
    1. Homogeneous Linear ODEs of Second Order
    2. Homogeneous Linear ODEs with Constant Coefficients
    3. Modeling of Free Oscillations of a Mass–Spring System
    4. Differential Operators.
    5. Euler–Cauchy Equations
    6. Existence and Uniqueness of Solutions.
    7. Nonhomogeneous ODEs
    8. Modeling: Forced Oscillations.
  • Linear Algebra and systems of ordinary differential equations
    1. Spaces and subspaces.
    2. Lineal combination and space generation.
    3. lineal dependence and lineal independece.
    4. Basis and dimension
    5. Linear transformations.
    6. Eigen-values and eigen-vectors
    1. Matrix: Operations and properties.
    2. Resolve linear systems by Gauss and Gauss-Jordan methods.
    3. Resolve linear systems by inverse and Cramer's rule.
    4. Applications: Leonlief product supplies analysis.
    1. systems of ordinary differential equations.
    2. Homogeneous systems of ordinary differential equations.
    3. Resolve linear systems of differential equations .
    4. linearization of high order differential equations.
  • Vectorial calculus. Several variable functions.
    1. Vectors in 2-Space and 3-Space.
    2. Inner Product (Dot Product).
    3. Vector Product (Cross Product).
    4. Vector and Scalar Functions and their Fields.
    5. Vector Calculus: Derivatives.
    6. Functions of Several Variables.
    7. Gradient of a Scalar Field.
    8. Directional Derivatives.
    9. Divergence of a Vector Field.
    10. Curl of a Vector Field.
    11. Lagrange multipliers.
  • partial differential equations.
    1. Introduction to partial differential equations (PDE)
    2. Solution by transforms methods

[edit] Week activities

Session dateTeacherTopicExternal resources
1 - 31-JulioTodos, Camilo
  • Presentation
  • Vectorial calculus. One variable calculus.
    1. Review of one variable calculus.
    2. Function definition.
    3. Derivatives.
    4. Integrals.
2 14 AgostoCamilo
3 21-AgostoAngelica
  • First order differential equations.
    1. Basic Concepts. Modeling.
    2. Geometric Meaning of y r ϭ f (x, y). Direction Fields, Euler’s Method.
    3. Separable ODEs. Modeling.
    4. Exact ODEs. Integrating Factors.
    5. Linear ODEs. Bernoulli Equation. Population Dynamics.
    6. Orthogonal Trajectories.
    7. Existence and Uniqueness of Solutions for Initial Value Problem.
4 28-AgostoAngélica
  • Homogeneous Linear ODEs of Second Order
  • Homogeneous Linear ODEs with Constant Coefficients
  • Modeling of Free Oscillations of a Mass–Spring System
  • Differential Operators.
  • Euler–Cauchy Equations
  • Existence and Uniqueness of Solutions.
  • Nonhomogeneous ODEs
  • Modeling: Forced Oscillations.
5 4-SeptJorge
  • Linear Algebra and systems of ordinary differential equations
    1. Spaces and subspaces.
    2. Lineal combination and space generation.
    3. lineal dependence and lineal independece.
    4. Basis and dimension
    5. Linear transformations.
    6. Eigen-values and eigen-vectors
6 11-SeptJorge
    1. Matrix: Operations and properties.
    2. Resolve linear systems by Gauss and Gauss-Jordan methods.
    3. Resolve linear systems by inverse and Cramer's rule.
    4. Applications: Leonlief product supplies analysis.
7 18-SeptAngélica
  • systems of ordinary differential equations.
  • Homogeneous systems of ordinary differential equations.
  • Resolve linear systems of differential equations .
  • linearization of high order differential equations.
8 25 -SeptJorge and Angélica
  • LAB: EDOs with MatLab, practice session.
9 2 -OctTodos
  • TEST
10 16 -OctCamilo
  1. Inner Product (Dot Product).
  2. Vector Product (Cross Product).
  • Vector and Scalar Functions and their Fields.
  1. Vector Calculus: Derivatives.
  2. Functions of Several Variables.
11 23 -OctCamilo

[edit] Versión en español

[edit] Objetivos

  • Fortalecer la formación matemática de los estudiantes
  • Proporcionar los fundamentos matemáticos necesarios para abordar adecuadamente las temáticas propias del modelado y simulación


[edit] Competencias generadas

[edit] Competencias interpretativas

  • Identificar las variables, constantes y parámetros que definen un sistema.
  • Leer, comprender e interpretar textos científicos con contenido matemático.
  • Asociar los resultados obtenidos a través del modelado con las características del sistema representado.
  • Expresar principios e hipótesis usando diferentes elementos del lenguaje matemático.

[edit] Competencias argumentativas

  • Establecer y analizar relaciones que representan fenómenos, sistemas y/o procesos.
  • Seleccionar y utilizar métodos apropiados para resolver problemas o sistemas.
  • Explicar ideas técnicas a través de textos, gráficas, ecuaciones e imágenes.

[edit] Competencias propositivas

  • Plantear un modelo matemático adecuado a casos particulares o problemas típicos.
  • Realizar diferentes tipos de representaciones para un único sistema.

[edit] Contenido

  1. SISTEMAS DE ECUACIONES LINEALES.
    1. Matrices: operaciones y propiedades
    2. Resolución de sistemas de ecuaciones por los métodos de Gauss y Gauss Jordan.
    3. Resolución de sistemas de ecuaciones lineales utilizando la inversa y regla de Cramer.
    4. Aplicaciones: Análisis de Insumo Producto de Leontief, Teoría de grafos y Cadenas de Markov.
      1. VECTORES EN Rn
        1. Vectores en Rn
        2. Operaciones entre vectores
        3. Producto punto, norma y proyecciones
        4. Producto cruz
        5. Aplicaciones: Paralelismo y ortogonalidad de vectores, vectores de área y de superficie.
          1. ESPACIOS VECTORIALES
            1. Espacios y sub-espacios
            2. Combinación lineal y espacio generado
            3. Dependencia e independencia lineal
            4. Bases y dimensión
            5. Transformaciones lineales.
            6. Valores propios y vectores propios
              1. FUNCIONES DE UNA Y VARIAS VARIABLES
                1. Generalidades de funciones en una variable
                2. Generalidades de funciones en varias variables
                3. Derivadas parciales, gradiente y derivadas direccionales
                4. Optimización
                5. Integración múltiple
                6. Análisis vectorial
                  1. ECUACIONES DIFERENCIALES
                    1. Ecuaciones diferenciales de primer orden
                    2. Ecuaciones de diferenciales de segundo orden
                    3. Sistemas de ecuaciones diferenciales
                    4. Solución mediante el método de transformadas
                    5. Introducción a las ecuaciones diferenciales parciales

                      6. [edit] Metodología

                      • La enseñanza de este curso se realizará a través de clases teóricas y prácticas en salas de cómputo.
                      • En el desarrollo de las clases prevalecerá la conceptualización en los temas a tratar sobre las destrezas operativas que pueden trabajarse mediante Sistemas Algebraicos Computacionales SAC.
                      • Durante el curso se hará énfasis en la importancia de cada tema en la formulación de modelos, considerando los diferentes campos de aplicación, más que en la implementación de algoritmos.

                      [edit] Bibliography/Bibliografía

                      1. Giordano, F., Fox, W., Horton, S., & Weir, M. (2009). 'A first course in mathematical modelling . Canada: Brooks/Cole Cengage Learning.
                      2. Grossman, S. (2008). Álgebra Lineal. México: Mc Graw Hill.
                      3. Kolman, B., & Hill, D. (2013). Álgebra Lineal. Fundamentos y apliaciones. Bogotá: Pearson.
                      4. Larson, R., & Edwards, B. H. (2010). Calculo 2 de Varias variables. China: Mc Graw Hill.
                      5. Meerschaaert, M. (2007). Mathematical Modeling. San Diego, United States of America: Elsevier Academic Press.
                      6. Zill, D. G. (2002). Ecuaciones diferenciales con aplicaciones de modelado. México: Cengage Learning.


                      [edit] Slides / Presentaciones

                      1. File:SemanaUno.pdf
                      2. File:SemanaDos.pdf
                      3. File:AlgLineal.pdf
                      4. File:Taller-de-ecuaciones.pdf

                      link Laboratorio sistemas.m

                      [edit] Matlab's Scripts

                      [edit] Linear dependency 

                       % PRUEBA PARA MOSTRAR INDEPENDENCIA LINEAL
                      
 x = rand(1, 3)-0.5;
                      
 y = rand(1, 3)-0.5;
                      
 a = rand(10, 1)-0.5;
                      
 b = rand(10, 1)-0.5;
                      
 z = a*x + b*y;
                      
 hold off
                      
 quiver3(0,0,0,x(1),x(2),x(3), 'r');
                      
 hold on
                      
 t= zeros(10,3);
                      
 quiver3(0,0,0,y(1),y(2),y(3), 'b');
                      
 quiver3(t(:,1),t(:,2),t(:,3),z(:,1),z(:,2),z(:,3),'g');
                      
 title('Dependencia lineal x rojo, y azul, combinaciones aleatorias verde ')

                      [edit] Lineal transformations 

                       %PRUEBAS CON EL DETERMINANTE EN MATRICES DE ESCALAMIENTO
 % construir el poligono con vertices x,y 
 x = [2 2 1 1 2 3 2  2  5 5 4 5 6 6 5 5 2];
 y = [1 4 4 7 7 8 9 11 11 9 8 7 7 4 4 1 1];
 x = x-3.5;
 y = y-5.5;
 hold off
 plot(x, y, 'b')
 title 'poligono original'
 axis([-7 7 -11 11])
 grid
 pause
 p = [x; y;];
 S1 = [1 0; 0 2]
 det_s1 = det(S1)
 S2 = [sqrt(2) 0; 0 sqrt(2)]
 det_s2 = det(S2)
 S3 = [0.5 0; 0 0.5]
 det_s3 = det(S3)
 S4 = S3*S2;
 det_s4 = det(S4)
 S5 = [0.5 0; 0 2]
 det_s5 = det(S5)
 p1 = S2*p;
 hold off
 plot(x, y, 'b:')
 hold on
 plot(p1(1, :), p1(2, :), 'r')
 title 'poligono escalado al doble'
 axis([-7 7 -11 11])
 grid
 pause
 p2 = S4*p;
 hold off
 plot(x, y, 'b:')
 hold on
 plot(p1(1, :), p1(2, :), 'b:')
 plot(p2(1, :), p2(2, :), 'r')
 title 'polígono escalado a la cuarta parte'
 axis([-7 7 -11 11])
 grid
 pause
 % PRUEBAS DE ACUMULAR LA ROTACION CON LA MULTIPLICACION 
 a = pi/4;
 R = [cos(a) -sin(a); sin(a) cos(a)]
 M = [1 0; 0 1];
 for i=1:8
   M = R*M;
   p1 = M*p;
   hold off
   plot(x, y, 'b:')
   hold on
   plot(p1(1, :), p1(2, :), 'r')
   title 'polígono rotacion acumulada'
   axis([-7 7 -11 11])
   grid
   pause	
 end
 % SINGULAR VALUE DECOMPOSITION PARA UNA MATRIZ GENERADA CON ROTACION Y ESCALAMIENTO
 R = [0.8 0.0 0.6; 0 1 0; -0.6 0.0 0.8]
 S = [0.5 0 0; 0 2 0; 0 0 1.5]
 M = S*R;
 [U, Z, V] = svd(M)
 % ANIMACION 3D CON UNA MATRIZ DE ROTACION APLICADA A PUNTOS EN UNA SUPERFICIE
 hold off 
 n = 400;
 x = 2*(rand(1, n)-0.5);
 y = 2*(rand(1, n)-0.5);
 z = x.^2 - y.^2;
 p = [x; y; z];
 for i=1:180,
   a = i*pi/90;
   rx = [1 0 0; 0 cos(a) -sin(a); 0 sin(a) cos(a)];
   pp = rx*p;
   plot(pp(1,:), pp(2,:), 'r.')
   axis([-1 1 -1 1])
   pause(0.1)
 end

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