Define multiplication on ℤ? by ? ⊙ ? = ? where ? is the remainder when the product ?? is divided by ?.

(a) Show that this operation is commutative.

(b) Show that this operation is associative.

(c) Show that there is an identity element for this operation.

(d) Show that for ? = 5, the set of non-zero elements forms a group under multiplication.

(e) Show that for ? = 6, the set of non-zero elements does not form under multiplication.

Use the method of variation of parameters to find a particular solution of the given differential equation and then find the general solution of the ODE.

y'' + y = tan(t)

If T : R 3 → R 3 is projection onto a line through the origin, describe geometrically the eigenvalues and eigenvectors of T.

In each of the following cases, give an explicit formula for s_2m.

(1) s_2n = 2s_n+s_n, s1 = 0

(2) s_2n = 2s_n-n, s1 = 3

(3) s_2n = 2s_n+5-7n, s1 = 0

Differential Equations

22. Solve each of the following systems of equations.

(c) (D-2)x=0; -3x+(D+2)y=0; -2y+(D-3)z=0

I got x=c₁e^2t, y= 3/4 C₁e^2t + C₂e^-2t by using diffieq method d/dx(ye^2t) = 3C₁e^2t....., but the right answer is

x=4c₁e^2t, y= 3C₁e^2t + 5C₂e^-2t , z= -6C₁e^2t -2C₂e^-2t+C3e^3t

I want to know where 4 came from, and please do not use matrix system.

V and W are finite dimensional inner product spaces,T: V→W is a linear map

1A: Give an example of a map T from R2 to itself (with the usual inner product) such that〈Tv,v〉= 0 for every map.

1B: Suppose that V is a complex space. Show that〈Tu,w〉=(1/4)(〈T(u+w),u+w〉−〈T(u−w),u−w〉)+(1/4)i(〈T(u+iw),u+iw〉−〈T(u−iw),u−iw〉

1C: Suppose T is a linear operator on a complex space such that〈Tv,v〉= 0 for all v. Show that T= 0 (i.e. that Tv=0 for all v).

2. For the curve y = 3x / x^2-1 determine each of the following and use all of the information to draw its graph

and notate the information on it.
(a) Find the domain of the function, as well as any x or y intercepts and symmetry.

(b) Find all vertical (2 of them) and horizontal (1) asymptotes. Support your work with limits.

(c) Use the first derivative to determine the maximums, minimums and intervals of increase/decrease of f(x).

(d) Use the second derivative to determine the points of inflection and intervals of concavity of f(x).

(e) Sketch the graph of f(x) and annotate it as follows: Sketch the asymptotes as dashed lines.

Plot and label the maxes (M), mins (m) inflection points (IP) on the graph.

Label the intervals on the graph for each of the 4 types of basic shapes:
1 for concave up/inc
2 for concave up/dec

3 for concave down/inc

4 for concave down/dec

Determine whether the equation is exact. If it is exact, find the solution. If it is not, enter NS. (8x^2−2xy+5)dx+(8y^2−x^2+8)dy=0

a.) Prove the following: Lemma. Let a and b be integers. If both a and b have the form 4k+1 (where k is an integer), then ab also has the form 4k+1.

b.)The lemma from part a generalizes two products of integers of the form 4k+1. State and prove the generalized lemma.

c.) Prove that any natural number of the form 4k+3 has a prime factor of the form 4k+3.

2. Let D be a relation on the natural numbers N defined by D = {(m,n) : m | n} (i.e., D(m,n) is true when n is divisible by m. For this problem, you’ll be proving that D is a partial order. This means that you’ll need to prove that it is reflexive, anti-symmetric, and transitive.

(a) Prove that D is reflexive. (Yes, you already did this problem on one of the minihomework assignments. You don’t have to redo the problem, but you should at least copy over your answer from that assignments to this one.)

(b) Prove that D is anti-symmetric. You may use the following fact: for any two natural numbers m and n, if m·n = 1, then m = 1 and n = 1. (Note that D is only anti-symmetric because the domain is the natural numbers. If we switched to the domain of integers, then things would be completely different.)

(c) Prove that D is transitive. (You’ve probably done a problem already that is almost exactly the same as this.)

how does gender impact the workplace and the home for both men and women? (300-500 words)

Show that the connected sum of two compact surfaces is a compact surface.

Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A countable?

Use the method of variation of parameters to determine a particular solution to the given equation.

y′′′+27y′′+243y′+729y=e^−9x

yp(x)=?