Expo Algebra Lineal

Elementary Linear AlgebraUVM/IIS

Thursday, July 8, 2010

EUCLIDEAN SPACE


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.


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


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


One DimensionR = R is the set of real numbers (i.e., the real line)

0-∞ ∞

0-∞ ∞1

√ 2

√ 2(1.41)

1


Two DimensionsR is called the Euclidean Space.

-∞ ∞

∞

0

∞

-∞

P(-2, 1)

2


Three Dimensions

-∞ ∞

∞

0

∞

-∞

P(2, 2, -2)y

x

z


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


SOLUTION OF EQUATIONS


Solutions of Systems of Linear Equations

-∞ ∞

∞

0

∞

-∞

x1 + x2 = 1

x1 - x2 = 1

x1 = 1

x2 = 0

HAS ONLY ONE SOLUTION:



-∞ ∞

∞

0

∞

-∞

x1 + x2 = 1

x1 + x2 = 2

HAS NO SOLUTIONS



-∞ ∞

∞

0

∞

-∞

x1 + x2 = 1

2x1 + 2x2 = 2

HAS INFINITELY MANY SOLUTIONS



∞

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.


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


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( )



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



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


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 = ...


Linear Algebra ApplicationGoogle PageRank


Early Search Engines

DATABASE OFWEB SITES

SEARCH QUERY

LIST OF MATCHING WEBSITESIN RANDOM ORDER

PROBLEM: HARD TO FIND USEFUL SEARCH RESULTS


Google Search Engine

DATABASE OFWEB SITES

SEARCH QUERY

MATCHING WEBSITESIMPORTANT SITES FIRST!

WITHRANKINGS!


How to Rank?

Ranking of a page = number of links pointing to that page

VERY SIMPLE RANKING:

PROBLEM: VERY EASY TO MANIPULATE


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


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:


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

( ( () ) )))=


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.


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