Expo Algebra Lineal
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Transcript of Expo Algebra Lineal
Elementary Linear AlgebraUVM/IIS
Thursday, July 8, 2010
EUCLIDEAN SPACE
Thursday, July 8, 2010
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
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).
It is commonly denoted R , although older literature uses the symbol E .
n
n
Thursday, July 8, 2010
Euclidean Space
R is a vector space and has Lebesgue covering dimension n.
Elements of R are called n-vectors.
R = R is the set of real numbers (i.e., the real line)
R is called the Euclidean Space.
n
n
1
2
Thursday, July 8, 2010
One DimensionR = R is the set of real numbers (i.e., the real line)
0-∞ ∞
0-∞ ∞1
√ 2
√ 2(1.41)
1
Thursday, July 8, 2010
Two DimensionsR is called the Euclidean Space.
-∞ ∞
∞
0
∞
-∞
P(-2, 1)
2
Thursday, July 8, 2010
Three Dimensions
-∞ ∞
∞
0
∞
-∞
P(2, 2, -2)y
x
z
Thursday, July 8, 2010
n Dimensions
R Space of One Dimension (x, y)
R Space of Two Dimensions (x, y)
R Space of Three Dimensions (x, y, z)
R Space of Four Dimensions (x1, x2, x3, x4)
R Space of n Dimensions (x1, x2, x3, ...., xn)
1
2
3
4
n
Thursday, July 8, 2010
SOLUTION OF EQUATIONS
Thursday, July 8, 2010
Solutions of Systems of Linear Equations
-∞ ∞
∞
0
∞
-∞
x1 + x2 = 1
x1 - x2 = 1
x1 = 1
x2 = 0
HAS ONLY ONE SOLUTION:
Thursday, July 8, 2010
Solutions of Systems of Linear Equations
-∞ ∞
∞
0
∞
-∞
x1 + x2 = 1
x1 + x2 = 2
HAS NO SOLUTIONS
Thursday, July 8, 2010
Solutions of Systems of Linear Equations
-∞ ∞
∞
0
∞
-∞
x1 + x2 = 1
2x1 + 2x2 = 2
HAS INFINITELY MANY SOLUTIONS
Thursday, July 8, 2010
Solutions of Systems of Linear Equations
∞
No solutions
Exactly one solution
Infinitely many solutions
A SYSTEM OF LINEAR EQUATIONS CAN HAVE EITHER:
In general:
Definition: If a system of equations has no solutions it is called an inconsistent system. Otherwise the system is consistent.
Thursday, July 8, 2010
Matrix NotationMATRIX = RECTANGULAR ARRAY OF NUMBERS
0 1 -2 4
2 0 0 1
1 1 3 9
EVERY SYSTEM OF LINEAR EQUATIONS CAN BE REPRESENTED BY A MATRIX
( () ) ))3 -1 1
2 0 2
Thursday, July 8, 2010
Elementary Row Operations
1. INTERCHANGE OF TWO ROWS
0 1 -2 4
2 0 0 1
1 1 3 9( ) ))1 1 3 9
2 0 0 1
0 1 -2 4( )
Thursday, July 8, 2010
Elementary Row Operations
2. MULTIPLICATION OF A ROW BY A NON-ZERO NUMBER
1 0 3 4
2 1 2 3
5 5 1 0( ) ))1 0 3 4
6 3 6 9
5 5 1 0( )*3
Thursday, July 8, 2010
Elementary Row Operations
3. ADDITION OF A MULTIPLE OF ONE ROW TO ANOTHER ROW
1 0 3 4
2 1 2 3
5 5 1 0( ) ))1 0 3 4
2 1 2 3
7 5 7 8( )*2
Thursday, July 8, 2010
How to Solve Systems of Linear Equations
))-1 2 3 4
2 0 6 9
4 -1 -3 0( )-x1 + 2x2 + 3x3 = 4
2x1 + 6x3 = 9
4x1 - x2 - 3x3 = 0
( )NICE MATRIX
x1 = ...
x2 = ...
x3 = ...
Thursday, July 8, 2010
Linear Algebra ApplicationGoogle PageRank
Thursday, July 8, 2010
Early Search Engines
DATABASE OFWEB SITES
SEARCH QUERY
LIST OF MATCHING WEBSITESIN RANDOM ORDER
PROBLEM: HARD TO FIND USEFUL SEARCH RESULTS
Thursday, July 8, 2010
Google Search Engine
DATABASE OFWEB SITES
SEARCH QUERY
MATCHING WEBSITESIMPORTANT SITES FIRST!
WITHRANKINGS!
Thursday, July 8, 2010
How to Rank?
Ranking of a page = number of links pointing to that page
VERY SIMPLE RANKING:
PROBLEM: VERY EASY TO MANIPULATE
Thursday, July 8, 2010
Google PageRank
Ranking of a page is x
The page has links to n other pages
IDEA: LINKS FROM HIGHLY RANKED PAGESSHOULD WORTH MORE
IF
THEN
Each link from that page should be worth x/n
Thursday, July 8, 2010
Google PageRank
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
THIS GIVES EQUATIONS:
Thursday, July 8, 2010
Google PageRank
MATRIX EQUATION:
x1
x2
x3
x4
0 0 1 1/2
1/3 0 0 0
1/3 1/2 0 1/2
1/3 1/2 0 0
COINCIDENCE MATRIXOF THE NETWORK
x1
x2
x3
x4
( ( () ) )))=
Thursday, July 8, 2010
Google PageRankx1
x2
x3
x4
0 0 1 1/2
1/3 0 0 0
1/3 1/2 0 1/2
1/3 1/2 0 0
x1
x2
x3
x4
( ( () ) )))=
( x1, x2, x3, x4 ) is an eigenvector of the coincidence matrix corresponding to the eigenvalue 1.
Thursday, July 8, 2010