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Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
Expo Algebra Lineal
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Expo Algebra Lineal

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Expo de FC para Algebra Lineal

Expo de FC para Algebra Lineal

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  • 1. Elementary Linear Algebra UVM/IIS Thursday, July 8, 2010
  • 2. EUCLIDEAN SPACE Thursday, July 8, 2010
  • 3. Euclidean Space is The Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity. Thursday, July 8, 2010
  • 4. Euclidean Space Euclidean n-space, sometimes called Cartesian space, or simply n-space, is the space of all n- tuples of real numbers (x1, x2, ..., xn). n It is commonly denoted R , although older n literature uses the symbol E . Thursday, July 8, 2010
  • 5. Euclidean Space n R is a vector space and has Lebesgue covering dimension n. n Elements of R are called n-vectors. R 1= R is the set of real numbers (i.e., the real line) 2 R is called the Euclidean Space. Thursday, July 8, 2010
  • 6. One Dimension 1 R = R is the set of real numbers (i.e., the real line) -∞ 0 ∞ √2 -∞ 0 1 √2 ∞ (1.41) Thursday, July 8, 2010
  • 7. Two Dimensions 2 R is called the Euclidean Space. ∞ P(-2, 1) -∞ 0 ∞ -∞ Thursday, July 8, 2010
  • 8. Three Dimensions y P(2, 2, -2) ∞ -∞ 0 ∞ x z -∞ Thursday, July 8, 2010
  • 9. n Dimensions 1 R Space of One Dimension (x, y) 2 R Space of Two Dimensions (x, y) 3 R Space of Three Dimensions (x, y, z) 4 R Space of Four Dimensions (x1, x2, x3, x4) n R Space of n Dimensions (x1, x2, x3, ...., xn) Thursday, July 8, 2010
  • 10. SOLUTION OF EQUATIONS Thursday, July 8, 2010
  • 11. Solutions of Systems of Linear Equations x1 + x 2 = 1 x1 - x 2 = 1 ∞ HAS ONLY ONE SOLUTION: x1 = 1 x2 = 0 0 -∞ ∞ -∞ Thursday, July 8, 2010
  • 12. Solutions of Systems of Linear Equations x1 + x 2 = 1 x1 + x 2 = 2 ∞ HAS NO SOLUTIONS -∞ 0 ∞ -∞ Thursday, July 8, 2010
  • 13. Solutions of Systems of Linear Equations x1 + x 2 = 1 2x1 + 2x2 = 2 ∞ HAS INFINITELY MANY SOLUTIONS -∞ 0 ∞ -∞ Thursday, July 8, 2010
  • 14. Solutions of Systems of Linear Equations In general: A SYSTEM OF LINEAR EQUATIONS CAN HAVE EITHER: No solutions Exactly one solution Infinitely many solutions Definition: If a system of equations has no solutions it is called an inconsistent system. Otherwise the system is consistent. Thursday, July 8, 2010
  • 15. Matrix Notation MATRIX = RECTANGULAR ARRAY OF NUMBERS ( )( ) ) 0 1 -2 4 3 -1 1 2 0 0 1 2 0 2 1 1 3 9 EVERY SYSTEM OF LINEAR EQUATIONS CAN BE REPRESENTED BY A MATRIX Thursday, July 8, 2010
  • 16. Elementary Row Operations 1. INTERCHANGE OF TWO ROWS ( )( ) ) 0 2 1 1 0 1 -2 0 3 4 1 9 1 2 0 1 0 1 3 0 -2 9 1 4 Thursday, July 8, 2010
  • 17. Elementary Row Operations 2. MULTIPLICATION OF A ROW BY A NON-ZERO NUMBER ( ) ( ) ) 1 2 5 0 1 5 3 2 1 4 3 0 *3 1 6 5 0 3 5 3 6 1 4 9 0 Thursday, July 8, 2010
  • 18. Elementary Row Operations 3. ADDITION OF A MULTIPLE OF ONE ROW TO ANOTHER ROW ( ) ( ) ) 1 2 5 0 1 5 3 2 1 4 3 0 *2 1 2 7 0 1 5 3 2 7 4 3 8 Thursday, July 8, 2010
  • 19. How to Solve Systems of Linear Equations ( ) -1 2 3 4 -x1 + 2x2 + 3x3 = 4 ) 2x1 + 6x3 = 9 2 0 6 9 4x1 - x2 - 3x3 = 0 4 -1 -3 0 ( ) x1 = ... x2 = ... NICE MATRIX x3 = ... Thursday, July 8, 2010
  • 20. Linear Algebra Application Google PageRank Thursday, July 8, 2010
  • 21. Early Search Engines SEARCH QUERY DATABASE OF WEB SITES LIST OF MATCHING WEBSITES IN RANDOM ORDER PROBLEM: HARD TO FIND USEFUL SEARCH RESULTS Thursday, July 8, 2010
  • 22. Google Search Engine DATABASE OF SEARCH QUERY WEB SITES WITH MATCHING WEBSITES RANKINGS! IMPORTANT SITES FIRST! Thursday, July 8, 2010
  • 23. How to Rank? VERY SIMPLE RANKING: Ranking of a page = number of links pointing to that page PROBLEM: VERY EASY TO MANIPULATE Thursday, July 8, 2010
  • 24. Google PageRank IDEA: LINKS FROM HIGHLY RANKED PAGES SHOULD WORTH MORE IF Ranking of a page is x The page has links to n other pages THEN Each link from that page should be worth x/n Thursday, July 8, 2010
  • 25. Google PageRank THIS GIVES EQUATIONS: x1 = x3 + 1/2 x4 x2 = 1/3 x1 x3 = 1/3 x1 + 1/2 x2 + 1/2 x4 x4 = 1/3 x1 + 1/2 x2 Thursday, July 8, 2010
  • 26. Google PageRank MATRIX EQUATION: ( ) ( )( ) ) x1 0 0 1 1/2 x1 x2 1/3 0 0 0 x2 = x3 1/3 1/2 0 1/2 x3 x4 1/3 1/2 0 0 x4 COINCIDENCE MATRIX OF THE NETWORK Thursday, July 8, 2010
  • 27. Google PageRank ( ) ( )( ) ) x1 0 0 1 1/2 x1 x2 1/3 0 0 0 x2 = x3 1/3 1/2 0 1/2 x3 x4 1/3 1/2 0 0 x4 ( x1, x2, x3, x4 ) is an eigenvector of the coincidence matrix corresponding to the eigenvalue 1. Thursday, July 8, 2010

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