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1 09:30 – 18:00 REGISTRATION 18:00 WELCOME RECEPTION PLACE: ISTANBUL UNIVERSITY GARDEN 09:00 – 10:00 OPENING CEREMONY 10:00 – 12:30 Chair : Metin ARIK 10:00 – 10:45 QUANTUM HALL EFFECT: A HOT TOPIC EVEN 25 YEARS AFTER THE NOBEL PRIZE Klaus von KLITZING – Max Planck Institute 10:45 – 11:00 Coffee Break 11:00 – 11:45 TURKISH ACCELERATOR CENTER PROJECT: THE STATUS AND ROAD MAP Ömer YAVAŞ – Ankara University 11:45 – 12:30 ELECTRONIC PROPERTIES OF QUANTUM DOTS: FROM CHAOS TO CONTROL Esa RÄSÄNEN – University of Jyvaskyla

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• Publié par: osso0
• Nombre de lectures: 7
• Langue: English

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Chapter 8
Linear Algebra
The subject of linear algebra includes the solution of linear equations,
a topic properly belonging to college algebra. The applied viewpoint
taken hereismotivated bythestudyofmechanicalsystemsandelectrical
networks, in which the notation and methods of linear algebra play an
important role.
Section 8.1 An introduction to linear equations requiring only a col-
lege algebra background: parametric solutions, reducedax+by = e
echelon systems, basis, nullity, rank and nullspace.cx+dy = f
Section 8.2 Matrix–vectornotation is introduced, especially designed
to prepare engineers and scientists to use computer userAX = b
interfaces from matlab and maple. Topics: matrix equa-0Y = AY
tions, change of variable, matrix multiplication, row oper-
ations, reduced row echelon form, matrix differential equa-
tion.
Section 8.3 Eigenanalysis for matrix equations. Applications to dif-
ferential equations. Topics: eigenanaysis, eigenvalue,AX = λX
eigenvector, eigenpair, ellipsoid and eigenanalysis, change−1P AP = D
of basis, diagonalization, uncoupled system of differential
equations, coupled systems.282 Linear Algebra
8.1 Linear Equations
Given numbers a , ..., a , b , ..., b and a list of unknowns x , x ,11 mn 1 m 1 2
..., x , consider the nonhomogeneous system of m linear equationsn
in n unknowns
a x +a x +···+a x = b ,11 1 12 2 1n n 1
a x +a x +···+a x = b ,21 1 22 2 2n n 2
(1) ...
a x +a x +···+a x = b .m1 1 m2 2 mn n m
Constants a , ..., a are called the coefficients of system (1). Con-11 mn
stants b , ..., b are collectively referenced as the right hand side,1 m
right side or RHS. Thehomogeneous system corresponding to sys-
tem (1) is obtained by replacing the right side by zero:
a x +a x +···+a x = 0,11 1 12 2 1n n
a x +a x +···+a x = 0,21 1 22 2 2n n
(2) ...
a x +a x +···+a x = 0.m1 1 m2 2 mn n
Anassignmentofpossiblevaluesx , ...,x whichsimultaneously satisfy1 n
all equations in (1) is called a solution of system (1). Solving system
(1) refers to the process of finding all possible solutions of (1). The
system (1) is called consistent if it has a solution and otherwise it is
called inconsistent.
In the plane (n = 2) and in 3-space (n = 3), equations (1) have a geo-
metric interpretation that can provide valuable intuition about possible
solutions. College algebra courses often omit the discussion of no so-
lutions or infinitely many solutions, discussing only the case of a single
unique solution. In contrast, all cases are considered here.
Plane Geometry. A straight line may be represented as an equa-
tion Ax+By = C. Solving system (1) is the geometrical equivalent of
finding all possible (x,y)-intersections of the lines represented in system
(1). The distinct geometrical possibliities appear in Figures 1–3.
y
Figure 1. Parallel lines, no solution.
−x+y = 1,
−x+y = 0.x8.1 Linear Equations 283
y
Figure 2. Identical lines, infinitely
many solutions.
−x+y = 1,
−2x+2y = 2.x
y
Figure 3. Non-parallel distinct lines,
one solution at the unique intersection
point P.
−x+y = 2,P
x x+y = 0.
Space Geometry. A plane in xyz-space is given by an equation
~Ax+By+Cz = D. The vector A~ı+B~+Ck is normal to the plane.
An equivalent equation is A(x−x )+B(y−y )+C(z−z ) = 0, where0 0 0
(x ,y ,z )isagivenpointintheplane. Solvingsystem(1)isthegeomet-0 0 0
ric equivalent of finding all possible (x,y,z)-intersections of the planes
represented by system (1). Illustrated in Figures 4–11 are some interest-
ing geometrical possibilities.
II P Figure 4. Knife cuts an open book.
Two non-parallel planes I, II meet in a line L
not parallel to plane III. There is a unique
point P of intersection of all three planes.
L I : y+z = 0, II : z = 0, III : x= 0.III
I
I
Figure 5. Triple–decker. Planes I, II, III
II are parallel. There is no intersection point.
III
I : z = 2, II : z = 1, III : z = 0.
I = II
Figure 6. Double–decker. Planes I, II
are equal and parallel to plane III. There is
III no intersection point.
I : 2z = 2, II : z = 1, III : z =0.284 Linear Algebra
Figure 7. Single–decker. Planes I, II, IIII = II = III
are equal. There are infinitely many
intersection points.
I : z = 1, II : 2z =2, III : 3z = 3.
I II
Figure 8. Pup tent. Two non-parallel
planes I, II meet in a line which never meets
III plane III. There are no intersection points.
I : y+z =0, II : y−z =0, III : z =−1.
I = II
Figure 9. Open book. Equal planes I, II
meet another plane III in a line L. There are
infinitely many intersection points.III
I : y+z =0, II : 2y+2z =0, III : z =0.
L
I III Figure 10. Book shelf. Two planes I, II
are distinct and parallel. There is no
intersection point.
II I : z =2, II : z =1, III : y = 0.
Figure 11. Saw tooth. Two non-parallelII
I planes I, II meet in a line L which lies in a
third plane III. There are infinitely many
intersection points.
III
I : −y+z = 0, II : y+z = 0, III : z = 0.
L
Parametric Solution. The geometric evidence of possible solution
sets gives rise to an algebraic problem:
What algebraic equations describe points, lines and planes?
The answer from analytic geometry appears in Table 1. In this table,
t and s are parameters, which means they are allowed to take on any
value between −∞ and +∞. The algebraic equations describing the
geometric objects are called parametric equations.8.1 Linear Equations 285
Table 1. Parametric equations with geometrical significance.
x = d , Point. The parametric equations describe a1
y = d , single point.2
z = d .3
x = d +a t, Line. The parametric equations describe a1 1
y = d +a t, straight line through (d ,d ,d ) with tangent2 2 1 2 3
~z = d +a t. vector a ~ı+a ~+a k.3 3 1 2 3
x = d +a s+b t, Plane. The parametric equations describe a1 1 1
y = d +a s+b t, planecontaining (d ,d ,d ). Thecrossproduct2 2 2 1 2 3
~ ~z = d +a s+b t. (a ~ı+a ~+a k)×(b ~ı+b ~+b k) is normal3 3 3 1 2 3 1 2 3
to the plane.
To illustrate, the parametric equations x = 2−6t, y = −1−t, z = 8t
describe the unique line of intersection of the three planes (details in
Example 1)
x + 2y + z = 0,
(3) 2x − 4y + z = 8,
3x − 2y + 2z = 8.
To describe all solutions of system (1), we generalize as follows.
Definition 1 (Parametric Equations, General Solution)
The terminology parametric equations refers to a set of equations of
the form
x = d +c t +···+c t ,1 1 11 1 1k k
x = d +c t +···+c t ,2 2 21 1 2k k
(4) ...
x = d +c t +···+c t .n n n1 1 nk k
Thenumbersd ,...,d ,c ,...,c areknown constantsandthevariable1 n 11 nk
names t ,...,t are parameters. A general solution or parametric1 k
solution of (1) is a set of parametric equations (4) plus two additional
requirements:
(5) Equations (4) satisfy (1) for−∞ < t <∞, 1≤ j≤ k.j
Any solution of (1) can be obtained from (4) by spe-
(6)
cializing values of the parameters.
Definition 2 (Minimal Parametric Solution)
Given system (1) has a parametric solution x , ..., x satisfying (4),1 n
(5), (6), then among all such parametric solutions there is one which
uses the fewest possible parameters. A parametric solution with fewest
parameters is called minimal, and otherwise redundant.286 Linear Algebra
Definition 3 (Gaussian Parametric Solution)
Parametric equations (4) are called Gaussian if they satisfy
(7) x = t , x = t , ..., x = ti 1 i 2 i k1 2 k
for distinct subscripts i , i , ..., i . The terminology is borrowed from1 2 k
Gaussian elimination, where such equations arise. A Gaussian para-
metricsolution ofsystem(1) isaset ofparametric equations(4) which
additionally satisfies (5), (6) and (7). See also equations (10), page 288.
For example, the plane x+y+z = 1 has a Gaussian parametric solution
x = 1−t −t ,y = t ,z = t ,whichisalsoaminimalparametricsolution.1 2 1 2
A redundant parametric solution of x+y+z = 1 is x = 1−t −t −2t ,1 2 3
y = t +t , z = t +t , using three parameters t , t , t .1 3 2 3 1 2 3
Theorem 1 (Gaussian Parametric Solutions)
A Gaussian parametric solution has the fewest possible parameters and it
represents each solution of the linear system by a unique set of parameter
values.
The theorem supplies the theoretical basis for the Gaussian algorithm to
follow (page 289), because the algorithm’s Gaussian parametric solution
isthenaminimalparametricsolution. TheproofofTheorem1isdelayed
until page 296. It is unlikely that this proof will be a subject of a class
lecture, due to its length; it is recommended reading after understanding
the examples.
Answer check algorithm. Although a given parametric solution
(4) can be tested for validity manually as in Example 2 infra, it is im-
portant to devise an answer check that free of parameters. The following
algorithm checksaparametricsolution bytesting constanttrial solutions
in systems (1) and (2).
Step 1. Set all parameters to zero to obtain the nonhomogeneous trial
solution x = d , x = d , ..., x = d . Test it by direct1 1 2 2 n n
substitution into the nonhomogeneous system (1).
Step 2. Consider the k homogeneous trial solutions
x = c , x = c , ..., x = c ,1 11 2 2