Use LU decomposition to solve the following system of equations (show your work). Do not use a pivoting strategy, and check your results by using the matrix inverse to show that [A][A]-1= [I].

8x+ 2y−z=10

- 2x+4y+z=5

3x−y+ 6z=7

Using a power series methodology, obtain the general solution (form u = c1u1 + c2u2 + f(x)) to the equation u” + 4u = x.

2. Consider a homogeneous system of linear equations with m equations and n variables.

   1. (i) Prove that this system is consistent.

   2. (ii) Prove that if m < n then the system has infinitely many solutions. Hint: Use r (the number of pivot columns) of the augmented matrix.

A voltage of 10 sin(3t) volts is impressed on a series circuit containing a 20Ω resistor, 10^-3 H inductor and a 1μF capacitor. Obtain expressions for the charge q on the capacitor and current I in the circuit if q=0=I at t=0.

Scenario 7.2 - Gulab Greatness
Historical demand for gulab jamun from a sweet stall on Commercial Road is as displayed in the table.

What is the trend component of Holt's model for period 0?

Key information:

Box dimension: 12” x 10” x 6” O.D.

Pallet dimensions: 48” x 42” x 6”

Constraints: (1) No pallet overhang

                    (2) Pallet unit load <= 48” high, including the pallet, for storage in a rack system

Trailer dimensions: 53’ long x 8’ 6” wide (b/w the hinges) x 9’ high

Deliverables:

1. Draw a 3-D sketch of the pallet unit load on a wood pallet. Include dimensions for a box and the pallet unit load.
2. Show your calculations for the quantity of boxes in a unit load.
3. Show your calculations for the number of pallets per trailer load.
4. Show your calculations for the quantity of boxes per trailer load.
5. Explain why these results are important to a producer who ships goods to customers by freightliner.

Let V = R^2×2 be the vector space of 2-by-2 matrices with real entries over
the scalar field R. We can define a function L on V by
L : V is sent to V
L = A maps to A^T ,
so that L is the “transpose operator.” The inner product of two matrices B in R^n×n and C in R^n×n is usually defined to be
<B,C> := trace (BC^T) ,
and we will use this as our inner product on V . Thus when we talk about
elements B,C in V being orthogonal, it means that <B,C> := trace (BC^T) = 0.
Problem 1.
1. First show that L is linear, so that L in B (V ).
2. Now choose a basis for the vector space V = R^2×2, and find the matrix of
L with respect to your basis.

Consider the initial value problem

y′ = 18x − 3y, y(0) = 2

(a) Solve it as a linear 1st order ODE with the method of the integrating factor.

(b) Solve it using a substitution method.

(c) Solve it using the Laplace transform.

Use the Runge-Kutta method with step sizes h = 0.1, to find approximate values of the solution of

y' + (1/x)y = (7/x^2) + 3 , y(1) = 3/2 at x = 0.5 .

And compare it to thee approximate value of y = (7lnx)/x + 3x/2

Consider a general system of linear equations with m equations in n variables, called system I. Let system II be the system obtained from system I by multiplying equation i by a nonzero real number c. Prove that system I and system II are equivalent.

(A) Using a straight edge, sketch the [1̅23̅] direction within a cubic unit cell and label start and end points as well as x, y, and z axes. (B) Using a straight edge, sketch the (1̅23̅) plane within a cubic unit cell and label x, y, and z axes as well as the x, y, z axial intercepts. (C) What is the geometric relationship between the (1̅23̅) plane and the [1̅23̅] direction?

solve the SDE dX = − 1 /τ( (X − µ)dt) + (2D)^-1/2 dB. X(t = 0) = β

Example 3.5: Again let X = Y = R. Define g by g(x) = x2. The graph of this function has the familiar parabolic shape as in Figure 3.1(b). Then for example, g([0, 1]) = [0, 1], g([1, 2]) = [1, 4], g({−1, 1}) = {1}, g−1([0, 1]) = [−1, 1], g−1([1, 2]) = [− √ 2, −1]∪[1, √ 2], g−1([0, ∞)) = R.

*I need help understanding why each example in bold is the answer it is*

*Please explain clearly why the inverse functions have the answer they have because it is not clear to me why*

*Please show all work and step by step solution*

Let G be a group. (consider the following parts that go together):

(1) Prove that (a^-1ba)ⁿ = a^-1bⁿa for any a,b in G, and any integer n.

(2) Prove that |xax^-1| = |a| for any a, x in G.

(3) Prove: If a is the only element of order two in G, then a lies in Z(G) where Z is the center of the group, G.

Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially there are 25 grams of A and 53 grams of B, and for each gram of B, 1.2 grams of A is used. It has been observed that 19.5 grams of C is formed in 15 minutes. How much is formed in 30 minutes? What is the limiting amount of C after a long time ?

_____ grams of C are formed in 30 minutes

_____ grams is the limiting amount of C after a long time