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11 Database Systems ( ) September 13, 2004 Lecture _1 By Hao-hua Chu ( ) 2 Course Goals • This is the first course in database management systems. • You will learn to: – Design relational databases – Use a relational database (SQL = Structured Query Language) – Build a relational database • This course will emphasize on hands-on learning: – A few programming assignments: build several components of a relational database system – A final project: build an online database application
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Outline of Lectures
in Linear Algebra Math 320
Spring 2012
Shmuel Friedland
University of Illinois at Chicago
http://www.math.uic.edu/∼friedlan
Lectures updated during the course
Last update January 31, 2012
11 Main Topics of the Course
• SYSTEMS OF EQUATIONS
• VECTOR SPACES
• LINEAR TRANSFORMATIONS
• DETERMINANTS
• INNER PRODUCT SPACES
• EIGENVALUES
• JORDAN CANONICAL FORM-RUDIMENTS
Text: Jim Hefferon, Linear Algebra, and Solutions
ftp://joshua.smcvt.edu/pub/hefferon/book/book.pdf
Software: MatLab,Maple, Matematica.
22 Applications of Linear Algebra
• Engineering
• Biology
• Medicine
• Statistics
• Physics
• Mathematics
• Numerical Analysis
Reason: Many real world systems consist of many parts
which interact linearly.
Analysis of such systems involves the notions and the tools
from Linear Algebra.
33 Lecture 1
I. Systems of Linear Equations
a x + a x + ... + a x =b11 1 12 2 1n n 1
a x + a x + ... + a x =b21 1 22 2 2n n 2
. . . . . . .
. . . . . . .
. . . . . . .
a x + a x + ... + a x =bm1 1 m2 2 mn n m
a. Examples
b. Solutions: Unique, Many and None (Inconsistent).
c. Graphical Examples of Systems in Two Variables
d. Equivalent Systems (have same solutions):
• Change the order of the equations
• Multiply an equation by a nonzero number
• Add (subtract) from one equation a multiple of another
equation
46
6
6
e. Triangular Systems and their solutions
a x + a x + ... + a x =b11 1 12 2 1n n 1
+ a x + ... + a x =b22 2 2n n 2
. . . . . . .
. . . . . . .
. . . . . . .
... a x =bnn n n
n equations inn unknowns withn pivots:
a =0, a =0,...a =0.11 22 nn
Solve the system by back substitution from down to up:
bn
x = ,n
ann
−a x +bn n−1(n−1)n
x = ,n−1
a(n−1)(n−1)
−a x −...−a x +bi+1 in n ii(i+1)
x = ,i
aii
i=n−2,...,1.
54 Lecture 1
II. Matrix Formalism for Solving Linear Equations
a. The Coefﬁcient Matrix of the system:
 
a a ... a11 12 1n
 
 a a ... a21 22 2n 
A= . . . . . . . .
. . . . 
a a ... am1 m2 mn
b. The Augmented Matrix(A|b), (A|B)
 
a a ... a | b11 12 1n 1
 
 a a ... a | b21 22 2n 2 
(A|b)= . . . . . . . . . .
. . . . | . 
a a ... a | bm1 m2 mn m
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6
5 Lecture 2
c. Elementary Row Operations (ERO)
• Interchange two rows
R ←→R , i=ji j
Example:R ←→R2 j
• Multiply a row by a nonzero number
a×R −→R , a=0, (R −→a×R ).i i i i
• Replace a row by its sum with a multiple of another row
R +a×R −→R , (R −→R +a×R ).i j i i i j
Example:
R−0.7R −→R , (R −→R−0.7R ).2 4 2 2 2 4
d. Pivotal Row
e. The elementary row operations are reversible: IfD is
obtained fromC using elementary row operations thenC is
obtained fromD using (the inverse elementary) row
operations
76
6
6 Inverse elementary row operation
R ←→R , i=j is inverse to itselfi j
1×R −→R , a=0i ia
is the inverse ofa×R −→Ri i
R −a×R −→Ri j i
is the inverse ofR +a×R −→Ri j i
−1
Denote byE the inverse elementary row operation
Assume thatD was obtained fromC by using the following
sequence ofk elementary row operations:
E E ...E Ek k−1 2 1
ThenC is obtained fromD by the elementary operations
−1 −1 −1 −1E E ...E E1 2 k−1 k
Elementary row operations on the system of linear equations
performed on augmented matrices give rise to the
equivalent system of equations
Two systems of linear equations are equivalent if they have
the same solutions
8Row Echelon Form of a matrix.
• The ﬁrst nonzero entry in each row is1. This entry is
called a pivot.
• If rowk does not consists entirely of zeros, then the
number of leading zero entries in rowk+1 is greater
then the number of leading zeros in rowk.
• Zero rows appear below the rows having nonzero
entries.
The process of using ERO to transform a linear system into
one whose augmented matrix is in row echelon form is
called Gaussian Elimination.
Corollary. The given system is inconsistent if and only if the
REF of its augmented matrix contains a row of the form:
[00 ...0|1] (6.1)
96
6
6
6
Examples of REF
 
1 a b c
 
 0 1 d e 
0 0 0 1
 
0 1 a b
 
 0 0 1 c 
0 0 0 0
Five possible REF of(abcd) (1×4 matrix):
(1uv w) if a=0,
(01pq) if a= 0, b=0,
(001r) if a=b=0, c=0,
(0001) if a=b=c=0, d=0,
(0000) if a=b=c=d=0.
10