Why is Gauss Elimination faster than solving a system of linear equations by using the inverse of a Matrix? (I know it has something to do with there being less operation with Gauss elim.) Can you show an example with a 2x2 and 3x3 matrix?

Let A and B be groups, and consider the product group G=A x B.

(a) Prove that N={(e_a,b) E A x B| b E B} is a subgroup.

(b) Prove that N is isomorphic to B

(c) Prove that N is a normal subgroup of G

(d) Prove that G|N is isomorphic to A

. Solve the Initial value problem by using Laplace transforms: ? ′′ + 3? ′ + 2? = 6? −? , ?(0) = 2 ? ′ (0) = 8

Find a basis and the dimension of W. Show algebraically how you found your answer.

a. W = {(x1, x2, x3, x4) ∈ R^4 | x2 = x3 and x1 + x4 = 0}

b. W = {( A ∈ M 3x3 (R) | A is an upper triangular matrix}

c. W = { f ∈ P3 (R) | f(0) = 0.

Show that if Y is a subspace of X, and A is a subset of Y, then the subspace topology on A as a subspace of Y is the same as the subspace topology on A as a subspace of X.

Prove or disprove. (If a proof seems difficult to finish, at least tell what you tried, or how far you got, or what seems to make the proof difficult.) (a) If a | b, then a 2 | b 2 . (b) If a 2 | b 2 , then a | b. (c) If a | b 2 , then a | b. (d) If (a, c) = 1 and (b, c) = 1, then (ab, c) = 1. (e) If (a, c) = 1 and (b, c) = 1, then (a, b) = 1.

Given X′=AX with X(t)=[x(t)y(t)], A=[23−4012−21] and X(0)=[3−4]. (a) Write the eigenvalues and eigenvectors of A λ1= , V1= ⎡⎣⎢⎢ ⎤⎦⎥⎥ , and λ2= , V2= ⎡⎣⎢⎢ ⎤⎦⎥⎥ (b) Write the solution of the initial-value problem in terms of x(t),y(t) x(t)= y(t)=

Show that the product of two Hausdorff spaces is Hausdorff.

(A) Prove division with remainder makes sense for integers as well as natural numbers. In other words prove the following.

Proposition: Let d be a nonzero integer. For any integer n, there exist unique integers q and r such that n = dq + r and 0 ≤ r < |d|.

Show that every order topology is Hausdorff.

For any two real sequences {an} and {bn}, prove that Rudin’s Ex. 5 We assume that the right hand side is defined, that is, not of the form ∞ − ∞ or −∞ + ∞.

lim sup (an + bn) ≤ lim sup an + lim sup bn.

Proof If lim sup an = ∞ or lim sup bn = ∞, there is nothing to prove

How do the three types of integer programming problems differ? Which do you think is most common, and why? Be detailed.

Prove that each element in Pentagon D5 has a unique inverse under the binary operation.

D5={AF, BF, CF, DF, EF,0,72,144,216,288}

Juan is a salesman for L. L. Bowers Corp. He has a choice of three compensation plans. Plan 1 pays $2500 per month. Plan 2 pays $2000 per month plus 15% commission. Plan 3 pays $1700 per month plus 30% commission. Graph the three plans and determine which is best. I need this graphed in excel but i am not sure how to do it.

The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model: C(x, y) = 1,000 + 200x2 + 200y2 where x is the reduction in sulfur emissions, y is the reduction in lead emissions (in pounds of pollutant per day), and C is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $100 per pound of sulfur and $200 per pound of lead removed. How many pounds of pollutant should the firm remove each day to minimize net cost (cost minus subsidy)?