first matrix A

[ 2 -1 3 ]

[-4 0 -2 ]

[2 -5 12 ]

[4 0 4 ]

amd b

[2]

[-2]

[5]

[0]

solve for Ax=b

using tan LU factorization of A

Let y′=y(4−ty) and y(0)=0.85.

Use Euler's method to find approximate values of the solution of the given initial value problem at t=0.5,1,1.5,2,2.5, and 3 with h=0.05.

Carry out all calculations exactly and round the final answers to six decimal places.

[ 1 -1 3 -3 5 2 ]

A=[ 1 -1 4 -1 9 -4 ]

[ -1 1 -3 3 -4 8  ]

[7]

b=[5]

[4]

use the row reduction algorithm to solve the following

Describe the solution set of Ax=b in parametric vector form

describe the solution set of Ax=0 as Span[ V1,V2,....,Vp]

Show that at least four of any 37 days must fall in the same month of the year

A study was conducted to determine whether the final grade of a student in an introductory psychology course is linearly related to his or her performance on the verbal ability test administered before college entrance. The verbal scores and final grades for 1010 students are shown in the table below.

Find the least squares line.

y=___ +___ x

III. Use Table to generate a list of ordered pairs (x,sin x) for x=0, \[Pi]/12, 2\[Pi]/12, 3\[Pi]/12, ..., 2\[Pi]. You should end up with a list of lists. Then do it again but without using Table.

Mathematica assignment, any ideas?

Fix a group G. We say that elements g₁, g₂∈G are conjugate if there exists h∈G such that

hg₁h⁻¹ = g₂.

1. Prove that conjugacy is an equivalence relation.
2. Prove that if g∈Z(G), the center of G, then its conjugacy classes has cardinality one.
3. Let G = S_n. Prove that h(i₁i₂ ... i_t)h⁻¹  = (h(i₁) h(i₂) ... h(i_t)), where i_j∈{1, 2, ... , n }.
4. Prove that the partition of S₃ into conjugacy classes is {{e} , {(1 2), (2 3), (1 3)} , {(1 2 3), (1 3 2)}} .That is, there are three distinct conjugacy classes: the set consisting of the 1-cycle e is one class, the set of 2-cycles is another class, and the set of 3-cycles forms the last conjugacy class.
5. Describe (with justification) the partition of S₄  into conjugacy classes explicitly. Be sure to be clear as to exactly how many conjugacy classes there are, give a representative element of each, and tell us how to determine which conjugacy class a given element of S₄ belongs. [Hint: You might want to invent a concept of "cycle type" to describe your answer.]
6. Are the elements

   1	2	3
   0	2	-7
   0	0	5

   and

   1	0	0
   0	5	π
   -1	0	2

   conjugate in the group GL₃(R)? Justify.

A recent 10-year study conducted by a research team at the Medical School was conducted to assess how age, blood pressure, and smoking relate to the risk of strokes. Assume that the following data are from a portion of this study. Risk is interpreted as the probability (times 100) that the patient will have a stroke over the next 10-year period. For the smoking variable, define a dummy variable with 1 indicating a smoker and 0 indicating a nonsmoker.

What action might the physician recommend for this patient?

Consider the mixing process shown in the figure. A mixing chamber initially contains 2 liters of a clear liquid. Clear liquid flows into the chamber at a rate of 10 liters per minute. A dye solution having a concentration of 0.75 kilograms per liter is injected into the mixing chamber at a constant rate of r liters per minute. When the mixing process is started, the well-stirred mixture is pumped from the chamber at a rate of 10+r liters per minute.

Part A and B provided. Please solve part C...

(a) Develop a mathematical model for the mixing process. Let Q represent the amount of dye in kilograms in the mixture.
dQ/dt = _______ kg / min

     3/4*r-Q/2(10+r)

(b) The objective is to obtain a dye concentration in the outflow mixture of 0.1 kilograms per liter. What injection rate r is required to achieve this equilibrium solution?
r =______ L / min

     20/13

Would this equilibrium value of r be different if the fluid in the chamber at time t=0 contained some dye?

(c) Assume the mixing chamber contains 2 liters of clear liquid at time t=0. How many minutes will it take for the outflow concentration to rise to within 1% of the desired concentration of 0.1 kilograms per liter?
t = ___________ min

prove that sign p=sign ^-p (if p is a permutation).

How many subgroups of order 9 and 49 may there be in a Group of order 441

A brine solution of salt flows at a constant rate of 4 ​L/min into a large tank that initially held 100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 3 ​L/min. If the concentration of salt in the brine entering the tank is 0.6 ​kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.1 ​kg/L?

Solve the Following Equation:

y'' + y' + y = a*sin(ω*t), y(0) = 0 , y'(0) = 0

Thanks

Describe the level surfaces for the 3-variable function: f(x,y,z) = z/(x-y)

Let p and q be any two distinct prime numbers and define the relation a R b on integers a,b by: a R b iff b-a is divisible by both p and q. For this relation R: Prove that R is an equivalence relation.

you may use the following lemma: If p is prime and p|mn, then p|m or p|n. Indicate in your proof the step(s) for which you invoke this lemma.