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 ∞
-∞
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8.
Three Dimensions
y
P(2, 2, -2)
∞
-∞ 0 ∞
x
z -∞
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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
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
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
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