Elementary Linear Algebra
                   UVM/IIS

Thursday, July 8, 2010
EUCLIDEAN SPACE
Thursday, July 8, 2010
Euclidean Space is
                    The Euclidean plane and three-dimensional space
                    of Euclidean ge...
Euclidean Space

                         Euclidean n-space, sometimes called Cartesian
                         space, or...
Euclidean Space

                         n
                    R is a vector space and has Lebesgue covering
            ...
One Dimension
                         1
                    R = R is the set of real numbers (i.e., the real line)



   ...
Two Dimensions
                         2
                    R is called the Euclidean Space.


                         ...
Three Dimensions

                                  y
                                             P(2, 2, -2)
           ...
n Dimensions
                         1
                    R Space of One Dimension (x, y)
                         2
   ...
SOLUTION OF EQUATIONS
Thursday, July 8, 2010
Solutions of Systems of
                   Linear Equations
               x1 + x 2 = 1
               x1 - x 2 = 1
      ...
Solutions of Systems of
                   Linear Equations
               x1 + x 2 = 1
               x1 + x 2 = 2
      ...
Solutions of Systems of
                   Linear Equations
               x1 + x 2 = 1
               2x1 + 2x2 = 2
     ...
Solutions of Systems of
                   Linear Equations
                         In general:

             A SYSTEM OF...
Matrix Notation
                         MATRIX = RECTANGULAR ARRAY OF NUMBERS




               ( )( ) )
               ...
Elementary Row
                   Operations
                                 1. INTERCHANGE OF TWO ROWS




        ( )( ...
Elementary Row
                   Operations
            2. MULTIPLICATION OF A ROW BY A NON-ZERO NUMBER




      ( ) ( )...
Elementary Row
                   Operations
            3. ADDITION OF A MULTIPLE OF ONE ROW TO ANOTHER ROW




      ( )...
How to Solve Systems
                   of Linear Equations

                                      (                      ...
Linear Algebra Application
                   Google PageRank

Thursday, July 8, 2010
Early Search Engines

                                  SEARCH QUERY
               DATABASE OF
                WEB SITES ...
Google Search Engine

               DATABASE OF       SEARCH QUERY
                WEB SITES
                    WITH    ...
How to Rank?

                               VERY SIMPLE RANKING:


                          Ranking of a page = number o...
Google PageRank
                              IDEA: LINKS FROM HIGHLY RANKED PAGES
                                       ...
Google PageRank
                           THIS GIVES EQUATIONS:



                         x1 = x3 + 1/2 x4
            ...
Google PageRank
                                  MATRIX EQUATION:




               ( ) ( )( ) )
                       ...
Google PageRank


                 ( ) ( )( ) )
                         x1             0    0   1   1/2        x1

      ...
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Expo Algebra Lineal

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Transcript of "Expo Algebra Lineal"

  1. 1. Elementary Linear Algebra UVM/IIS Thursday, July 8, 2010
  2. 2. EUCLIDEAN SPACE Thursday, July 8, 2010
  3. 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. 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. 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. 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. 7. Two Dimensions 2 R is called the Euclidean Space. ∞ P(-2, 1) -∞ 0 ∞ -∞ Thursday, July 8, 2010
  8. 8. Three Dimensions y P(2, 2, -2) ∞ -∞ 0 ∞ x z -∞ Thursday, July 8, 2010
  9. 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. 10. SOLUTION OF EQUATIONS Thursday, July 8, 2010
  11. 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. 12. Solutions of Systems of Linear Equations x1 + x 2 = 1 x1 + x 2 = 2 ∞ HAS NO SOLUTIONS -∞ 0 ∞ -∞ Thursday, July 8, 2010
  13. 13. Solutions of Systems of Linear Equations x1 + x 2 = 1 2x1 + 2x2 = 2 ∞ HAS INFINITELY MANY SOLUTIONS -∞ 0 ∞ -∞ Thursday, July 8, 2010
  14. 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. 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. 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. 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. 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. 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. 20. Linear Algebra Application Google PageRank Thursday, July 8, 2010
  21. 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. 22. Google Search Engine DATABASE OF SEARCH QUERY WEB SITES WITH MATCHING WEBSITES RANKINGS! IMPORTANT SITES FIRST! Thursday, July 8, 2010
  23. 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. 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. 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. 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. 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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