True/False: If A is an invertible m x m matrix and B is an m x n matrix, then AB and B have the same null space.

(Hint: AB and B have the same null space means that ABX = 0 if and only if BX = 0, where X is in Rⁿ.)

Course: Differential Geometry (Vector Calculus & Linear Algebra)

Provide all proofs

(a) Find the formula for the distance from p to the line y=mx

(b) prove that the set U={(x,y): y<mx} is an open set

Prove the following theorem:

In a Pasch geometry, a quadrilateral is a convex quadrilateral if and only if the vertex of each angle is contained in the interior of the opposite angle.

Problem 2-25 (Algorithmic)

George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 8% for the bond fund and 20% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 40% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 5.5%.

1. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives. If required, round your answers to three decimal places.
   

   Let B	=	percentage of funds invested in the bond fund
   S	=	percentage of funds invested in the stock fund

   	B	+	S
   s.t.
   	B					Bond fund minimum
   	B	+	S			Minimum return
   	B	+	S			Percentage requirement

2. Solve the problem. If required, round the answers to one decimal place.
   
   Optimal solution: B = , S =
   
   Value of optimal solution is  % ????

A rectangular box with no top is to have a surface area of

64 m².  Find the dimensions (in m) that maximize its volume.

I got width as X= 8sqrt3/3, length y=8sqrt3/3, and height as z=4sqr3/3 but it is wrong and I don't know why

An eulerian walk is a sequence of vertices in a graph such that every edge is traversed exactly once. It differs from an eulerian circuit in that the starting and ending vertex don’t have to be the same. Prove that if a graph is connected and has at most two vertices of odd degree, then it has an eulerian walk.

Compute φ(21). Find all the units modulo 21 and write down their inverses.

Find all RING homomorphisms from Z18 to Z18

we are going to focus on basic logic and how you can use logic outside of the classroom. Unfortunately,

many of our daily interactions include logical errors.

Respond to the following prompts in a minimum of 175 words:

• Consider a recent interaction you have had with a co-worker or item that you saw in the media that might have included logical errors.
• Share the example as well as the logical errors that are present in the example. Be sure to use terms and concepts from the content covered this week to support your discussion.

Solve the differential equation using the Laplace transform.   y''' + 3y''+2y' = 100e^-t , y(0) = 0, y'(0) = 0, y''(0) = 0

What is the number of permutations of the six digits 123456 in which the strings 123, 234, 345, 456, 654, 543, 432, 321 don't appear?

1) Find y as a function of t if 9y′′+24y′+32y=0,

y(0)=5,y′(0)=8. y(t)=

2) Find y as a function of x if y′′′+16y′=0,

y(0)=−5,  y′(0)=−32,  y′′(0)=−32. y(x)=

3) Find y as a function of t if 9y′′−12y′+40y=0,

y(1)=5,y′(1)=9. y=

What is overmatching versus undermatching? Give an example of overmatching and undermatching (with hypothetical proportions) that might occur on a concurrent VI 20-sec VI 30-sec schedule (which means that you must first work out the expected proportions with these schedules).

JOAN+CAN=START

Solve this addition problem by substituting numbers by the given letters, to find the value of start. Each letter represents a unique digit