Mu Alpha Theta National Convention: Denver, 2001
Geometry Topic Test Solutions Alpha Division
Mu Alpha Theta National Convention: Denver, 2001
Geometry Topic Test Solutions Alpha Division
Mu Alpha Theta National Convention: Denver, 2001
Geometry Topic Tes
Homework 7 Solutions
Problem 1 (6.4.18). Let T : V W where V, W are finite dimensional inner product spaces. Show that (a) (2 points) T T and T T are positive semi-definite, and (b) (4 points) rank(T ) = rank(T T ) = rank(T T ). Proof. (a) Clearly T T is
Throughout in this text V will be a vector space of nite dimension n
over a eld K and T : V V will be a linear transformation.
1
Eigenvalues and Eigenvectors
A scalar K is an eigenvalue of T if there is a nonzero v V such that
T v = v. In this case v is c
1
Review
Last lecture we learnt how to orthogonally diagonalise a real symmetric matrix A.
Quadratic Forms: Conic Sections
2
The equation: ax 2 + bxy + cy 2 + dx + ey + f = 0 can always be
written in matrix form:
x
y
a
b/2
b/2
c
x
+d
y
e
x
+f =0
y
or as x
16
Notes on Jordan-Hlder
o
Denition 16.1. A normal series of a group G is a sequence of subgroups:
G = H0 H1 H2 Hm = 1
so that each subgroup is normal in the previous one (Hi Hi1 ). The quotient
groups Hi1 /Hi is called a subquotients. A renement of a nor
MATH 30A FALL 04
Homework This is homework from Math 30a, Fall 2004. I was teaching only one course. So I had a lot of time to prepare the course. The book was John Fraleigh A rst course in Abstract Algebra, 7th edition. We covered the rst four chapters.
4.3-QUS-12
S8:
An easy element of order 4 is (1234), so how to get others?
We can put two disjoint 4 cycles together: (1234)(5678)
We can also add a disjoint 2 cycles: (1234)(56), (1234)(56)(78)
That's it. Any cycle length divides the order of the el
(January 14, 2009)
[16.1] Let p be the smallest prime dividing the order of a nite group G. Show that a subgroup H of G of
index p is necessarily normal.
Let G act on cosets gH of H by left multiplication. This gives a homomorphism f of G to the group of
GROUPS: SUPPLEMENTARY TOPICS
Theorem (see Proposition 2.59, page 172):
If G is a nite abelian group, then G has a subgroup of order d for every divisor d of |G|.
Theorem (Cayley) (see Theorem 2.66, page 178):
Every group G is isomorphic to a subgroup of t
7.3.12. If G is a nite group of order n, and p is the least prime such
that p j n, show that any subgroup of index p is normal in G.
Let H be a subgroup of index p in G. Consider the action of G on the set
S of left cosets of H given by a xH = axH as in E
Review of the Method of Differentiation Notes by C. Ho
I The Basic Cases HB - oe ! HB sin B oe cos B HB sec B oe sec B tan B HB /B oe /B II Algebraic Combinations HB -0 B oe -HB 0 B HB 0 B , 1B oe HB 0 B , HB 1B HB 0 B 1B oe HB 0 B 1B 0 B HB 1B
Draw a 8catte1 plot of score V ~ U time treating time as the independent variable. Then use two points to make an B equation f d a line of best-fit. Skaeh the line on your edtter plot. Use the equation to estimate the score a ~tudent mtght d t e a f
Math 072 Study Guide for Test 2 (6th Ed.), Wednesday, April 2
Ask me if you need any help. My email is [email protected] Techniques of Integration (50%) This is the main topic for this test. Given an integral, there are certain things you should as
Math 072 Study Guide for Test 1
(Wednesday, February 27)
The following topics will be covered. Review the concepts, the examples given in class, and the problems assigned. Make sure that you know how to compute derivatives and integrals (especially
Math 072 Test 1, Spring 2008 Name: Solution $ 1. Let 0 B oe B 5B -9= B % (5 pts) a. Show that 0 has an inverse on the real line You need to give an analytic argument, not just rely on graphs (graphs are hard to be accurate). Ans. Since 0 w B oe $B