Expo Algebra Lineal

Elementary Linear Algebra
UVM/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).
n
It is commonly denoted R , although older
n
literature uses the symbol E .


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.


One Dimension
1
R = R is the set of real numbers (i.e., the real line)

-∞ 0 ∞

√2

-∞ 0 1 √2 ∞
(1.41)


Two Dimensions
2
R is called the Euclidean Space.

∞

P(-2, 1)

-∞ 0 ∞

-∞


Three Dimensions

y
P(2, 2, -2)
∞

-∞ 0 ∞
x

z -∞


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)


SOLUTION OF EQUATIONS

Solutions of Systems of
Linear Equations
x1 + x 2 = 1
x1 - x 2 = 1
∞
HAS ONLY ONE SOLUTION:

x1 = 1
x2 = 0 0
-∞ ∞

-∞


Linear Equations
x1 + x 2 = 1
x1 + x 2 = 2
∞
HAS NO SOLUTIONS

-∞ 0 ∞

-∞


Linear Equations
x1 + x 2 = 1
2x1 + 2x2 = 2
∞
HAS INFINITELY MANY
SOLUTIONS

-∞ 0 ∞

-∞


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.


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

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


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


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


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 = ...


Linear Algebra Application
Google PageRank


Early Search Engines

SEARCH QUERY
DATABASE OF
WEB SITES LIST OF MATCHING WEBSITES
IN RANDOM ORDER

PROBLEM:
HARD TO FIND USEFUL SEARCH RESULTS


Google Search Engine

DATABASE OF SEARCH QUERY
WEB SITES
WITH MATCHING WEBSITES
RANKINGS! IMPORTANT SITES FIRST!


How to Rank?

VERY SIMPLE RANKING:

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

PROBLEM: VERY EASY TO MANIPULATE


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


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


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

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.


Expo Algebra Lineal

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