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 The
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
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
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
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
group
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 A
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)
(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 multiplica
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, pa
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 th
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
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 equatio
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 a
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 ho
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 (graph