Math 660 GEOMETRY I
SOLUTION FOR QUIZ IX (11/26)
December 3 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (20pts)
Let
1 : 1 : 0 : 0 ,
2 : 0 : 1 : 0 ,
3:0:0:1
be three points inside P3 (X :Y :Z :W ) .
(1)
The answer is False. These three points
Math 660 GEOMETRY I
CRASH COURSE ON THE COMPLEX NUMBER FIELD I
December 3 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
31. Introduction to Complex Numbers. Arithmetic. Commutativity.
As you might realize by now, one of the main focuses in project
Math 660 GEOMETRY I
FINAL EXAM (Take-home)
December 10 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
ID # :
Name :
This take-home part of Final Exam is worth 200 points and is
due Monday, December 16th, 2013. Submission after 5 : 00 pm,
December 1
Math 660 GEOMETRY I
REVIEW OF LECTURES II
August 26 (Wed), 2015
Instructor:
Yasuyuki Kachi
Line #: 25941.
2. Models.
Today, I want to talk about models. Models, as in geometric models. Not
supermodels. No Secret of Queen Victoria. The notion of models act
Math 660 GEOMETRY I
REVIEW OF LECTURES III
August 31 (Mon), 2015
Instructor:
Yasuyuki Kachi
Line #: 25941.
3. Projective geometry.
How are you, everyone? Last time I told you there is actually a plane wherein two
lines always intersect. That sure sounds o
Math 660 GEOMETRY I
FINAL EXAM (Take-home)
December 10 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
ID # :
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Name :
This take-home part of Final Exam is worth 200 points and is
due Mon
Math 660 GEOMETRY I
SOLUTION FOR QUIZ VI (10/19)
October 26 (Mon), 2015
Instructor:
Yasuyuki Kachi
Line #: 25941.
[I] (10pts)
(1)
The equation of the tangent line to the curve
XY + 3 Y Z Z 2 = 0
inside P2 (X:Y :Z) at
X : Y : Z
=
2 : 1 : 2
is found as foll
Math 660 GEOMETRY I
REVIEW OF LECTURES VI
September 14 (Mon), 2015
Instructor:
Yasuyuki Kachi
Line #: 25941.
6. Intersection of two lines in P2 .
Intersection of two lines.
Here we go, finally:
In P2 , endowed with a homogeneous coordinate
X : Y
: Z ,
it
Math 660 GEOMETRY I
REVIEW OF LECTURES IX
September 16 (Wed), 2015
Instructor:
Yasuyuki Kachi
Line #: 25941.
9. Projective 3-space.
Projective 3-space.
Today I would like to show you how the picture of a projective plane is carried
over to its higher-dim
Math 660 GEOMETRY I
REVIEW OF LECTURES VII
September 14 (Mon), 2015
Instructor:
Yasuyuki Kachi
Line #: 25941.
7. Duality.
Duality.
Last time we have proved that two lines always intersect in P2 . Today, we will
cover the other featured topic: How the one
Math 660 MODERN GEOMETRY I
REVIEW OF LECTURES IV
September 2 (Wed), 2015
Yasuyuki Kachi
Instructor:
Line #: 25941.
4. Homogeneous coordinate.
Last time we were examining how a line in an xy-coordinate plane designates
a ratio of three real numbers. Let me
Math 660 GEOMETRY I
REVIEW OF LECTURES V
September 9 (Wed), 2015
Instructor:
Yasuyuki Kachi
Line #: 25941.
5. P2 as a completion of an xy-plane.
We are in the process of building a geometry where a ratio serves as a point .
In that geometry, the notion of
Math 660 GEOMETRY I
REVIEW OF LECTURES VIII
September 16 (Wed), 2015
Instructor:
Yasuyuki Kachi
Line #: 25941.
8. Incidence.
Incidence.
Now we have addressed two major questions that naturally arose in our introductory discussion what is geometry?
1. The
Math 660 GEOMETRY I
REVIEW OF LECTURES XXIX
November 21 (Thu), 2013
Yasuyuki Kachi
Instructor:
Line #: 26471.
29. Linear span III.
Now it is almost clear how to generalize the notion a linear span for k distinct
points residing in Pn .
Assumption 1.
Let k
Math 660 GEOMETRY I
REVIEW OF LECTURES XXX
November 26 (Tue), 2013
Yasuyuki Kachi
Instructor:
Line #: 26471.
30. Linear span IV.
Lets review the formal denition of the linear span from last time:
Assumption 1.
Let k and n be positive integers, such that k
Math 660 GEOMETRY I
SOLUTION FOR REGULAR HOMEWORK III (10/03)
October 10 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I]
(8pts)
(1)
The system
(#)
True or false :
a1 X + b1 Y + c1 Z + d1 W = 0,
a2 X + b2 Y + c2 Z + d2 W = 0
denes a line in P3 (X
Math 660 GEOMETRY I
SOLUTION FOR MIDTERM EXAM I (In-class; 10/22)
October 22 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (20pts)
(1)
Let
y, z
be the ane coordinate of A2 inside P2 (X :Y :Z ) ,
with
y=
Then the de-homogenization of
y, z
(2)
L
Math 660 GEOMETRY I
SOLUTION FOR QUIZ I (09/12)
September 17 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (4pts)
(1)
Homogenization of
X:Y :Z
(2)
=
=
1:
:Z
x , z
Homogenization of
1
,
2
=
X:Y :Z
[II] (4pts)
(1)
with respect to Z is
(2)
=
X:Y
Math 660 GEOMETRY I
SOLUTION FOR QUIZ II (09/24)
September 26 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (20pts)
(1)
In P2 (X :Y :Z ) :
Identify the point of intersection of the two lines
4 X + Y Z = 0,
2X + 2Y + 3Z
=
0.
Solution for (1) :
Math 660 GEOMETRY I
SOLUTION FOR QUIZ III (09/24)
September 26 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (20pts)
(1)
In P2 (X :Y :Z ) :
The dening equation of the line that passes through
4 : 1 : 1
and
2:2:3
is
5 X 14 Y + 6 Z = 0.
(2)
The
Math 660 GEOMETRY I
SOLUTION FOR QUIZ IV (10/03)
October 8 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (20pts)
In P3 (X :Y :Z :W ) :
1 :
4X 2Y +
(1)
X+
7X +
Z 3W
Y + 2Z
= 0,
W
= 0,
Y + 7Z 6W
= 0,
and
2 :
coincide . Indeed,
P l 1 =
=
P l 2 =
Math 660 GEOMETRY I
SOLUTION FOR QUIZ V (10/10)
October 17 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (20pts)
(1)
No,
P0 : P1 : P2 : P3 : P4 : P5
=
1 : 2 : 3 : 4 : 1 : 7
does not arise as the Plcker coordinate of any line P3 (X :Y :Z :W ) .
Math 660 GEOMETRY I
SOLUTION FOR QUIZ VI (10/17)
October 22 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (10pts)
(1)
The equation of the tangent line to the curve
XY + 3 Y Z Z 2 = 0
inside P2 (X :Y :Z ) at
X:Y :Z
=
2 : 1 : 2
is found as follo
Math 660 GEOMETRY I
SOLUTION FOR QUIZ VII (10/29)
October 31 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (20pts)
(1) The two lines and , both residing in P3 (X :Y :Z :W ) , whose
Plcker coordinates are
u
Pl =
1:2:3:0:1:2,
P l
(a) do not in
Math 660 GEOMETRY I
SOLUTION FOR QUIZ VIII (10/31)
November 5 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
[I] (15pts)
(1)
True or false :
In P3 , two lines always intersect.
The answer is false .
Reason:
For two lines and in P3 ,
codimP3 = 3 1
Math 660 GEOMETRY I
REVIEW OF LECTURES XXVII
November 19 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
27. Linear span I.
The end of the semester is approaching. It was quite a journey. At the
beginning of the semester I said there is a geometry w
Math 660 GEOMETRY I
REVIEW OF LECTURES XXVIII
November 21 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 26471.
28. Linear span II.
Last time we covered the notion of a linear span for three distinct points residing
in Pn that are non-collinear, with n 3.