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?
In: Advanced Math
Let A and B be groups, and consider the product group G=A x B.
(a) Prove that N={(ea,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
In: Advanced Math
. Solve the Initial value problem by using Laplace transforms: ? ′′ + 3? ′ + 2? = 6? −? , ?(0) = 2 ? ′ (0) = 8
In: Advanced Math
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.
In: Advanced Math
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.
In: Advanced Math
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.
In: Advanced Math
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)=
In: Advanced Math
Show that the product of two Hausdorff spaces is Hausdorff.
In: Advanced Math
(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|.
In: Advanced Math
Show that every order topology is Hausdorff.
In: Advanced Math
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
In: Advanced Math
How do the three types of integer programming problems differ? Which do you think is most common, and why? Be detailed.
In: Advanced Math
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}
In: Advanced Math
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.
In: Advanced Math
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)?
In: Advanced Math