19 September 2015
Today I took the Counseling 200A class at 8am,although I already know some of the
thing that teacher told in the class, I still get three important thing which will helps me
His Life and His Faith
Dr. George W Benthien
November 8, 2008
E-mail: [email protected]
Leonhard Euler (17071783)
When I was a mathematics student in college there were many times when I encountered the name
Euler Euler eq
Proofs involving Normal Subgroups
December 8, 2009
Let G be a group. The set Z(G) = cfw_x G|xg = gx for all g G of all elements
that commute with every other element of G is called the center of G. Prove
that Z(G) is a normal subgroup of G.
The center has
Whats new in this version?:
In the rst version, the basis for question (i) were not orthonormalized, thus making the algebra done in the orthogonal
decomposition more scary than it is. We xed this by making the basis for M orthonormal. For question (ii),
Since (xn ) is Cauchy, then let
> 0 such that when n, nk > N1 implies d(xnk , xn ) < . Similiarly, since (xnk ) converges
to x, let > 0 and when nk > N1 implies d(xnk , x) < .
Set n = maxcfw_N1 , N2 , and we have
d(xn , x) d(xnk ,
Let x, y B(0; 1), then x 1 and y 1. We show z B(0; 1), thus
x + (1 )y x + (1 ) y + 1 = 1
Every norm is a convex function, we will show that
(x + (1 )y) (x) + (1 )(y)
is not true.
Proof On the contrary, assume (1) is true, that is giv
Assignments for week 2: History 7A
Prof. Story: Summer 2016:
Student Choice: Pick 3 out of the following 6 assignments.
Assignments 1 to 5 can be completed by submitting a hard copy of
Assignment #6 must be completed online.
1. Fill out a ch
Assignments for week 1: History 7A
There have been problems with the Moodle shell this week so I am
not requiring you to do any online work this week. You may turn in
hard copies of the following assignments and you do not need to turn
them in online.
Exam 3 Solutions
1. Let A =
a) What is the characteristic polynomial chA ()?
= ( 3)2 .
Solution: chA () =
b) For each eigenvalue of A, find a basis for the associated eigenspace.
Solution: The only eigenvalue