Explain how you see the relationship between the three elementary row operations performed on an augmented matrix and the operations that lead to equivalent systems of equations. What advantages do you see in converting a system of equations to an equivalent augmented matrix? Research a particular application of matrices explaining how a matrix is used. Based on what you have selected think about the three operations of addition, scalar multiplication, and matrix multiplication. Can you describe what the results actually mean? Note that in some applications some of these operations may not have meaning. If that is the case, explain why.

8.Let a and b be integers and d a positive integer.
(a) Prove that if d divides a and d divides b, then d divides both a + b and a − b.
(b) Is the converse of the above true? If so, prove it. If not, give a specific example of a, b, d showing
that the converse is false.

9. Let a, b, c, m, n be integers. Prove that if a divides each of b and c, then a divides nb + mc.

An environmentally toxic radioactive chemical is continually released at a constant rate of 1 (mg per vol per time) at the midpoint of a canal of length L with still water. As it diffuses through the canal with diffusion constant D = 1, it decays at rate λ (per unit time) The ends of the canal are connected to large bodies of toxic-free water. Set up the model equations and the boundary conditions. Find the steady-state concentration and sketch its spatial profile for different values of L and λ.

[Hint: Break the problem up into two parts, on each side of the source. At the source the concentration must be continuous, and in a small interval about the source, the ‘flux in’ minus the ‘flux out’ equals one.]

#1 Let H= Span{v1,v2,v3,v4}. For each of the following sets of vectors determine whether H is a line, plane ,or R3. Justify your answers.

(a)v1= (1,2,−2),v2= (7,−7,−7),v3= (16,−12,−16),v4= (0,−3,−3)

(b)v1= (2,2,2),v2= (6,6,5),v3= (−16,−16,−14),v4= (28,28,24)

(c)v1= (−1,3,−3),v2= (0,0,0),v3= (−2,6,−6),v4= (−3,9,−9)

#2 Plot the linesL1: x= t[4−1] and L2: x= [−4−2] + t[4−1] using their vector forms. If[12k]is onL2. What is the value of k?

Formulate Newton’s method for the system
x^3 = y

y^3 = x

Note that this system has three real roots (−1, −1), (0, 0) and (1, 1). By taking various initial points in the square −2 < x, y < 2, try to obtain the attraction regions of these three roots.

P.S. An attraction region of a root is defined as the set of all initial points which will eventually converge to the root.

For the following, you can either provide a vector equation for the curve, or you can describe the curve in words with sufficient details.

a) Describe a curve that has curvature zero.
b) Describe a curve that has torsion zero.
c) Describe a curve that has constant nonzero curvature.
d) Describe a curve that has constant nonzero torsion.
e) Describe a curve that has zero curvature and zero torsion.

Q. I would like you to use a fixed point method to solve the positive real quadratic root of 4 by solving h(x) = x^4 − 4 = 0. The standard method manipulates h(x) = 0 into g(x) = x so that the iterative scheme becomes xn+1 = g(xn). The iterative scheme will converge to the required solution if the root is in the interval defined by |g '(x)| < 1

(i) We begin by adding x to both sides of the equation to form x = x^ 3− 4+ x such that g(x) = x ^3 + x − 4. Will the root be found? (ii) We now try adding −2x to both sides of the equation to form −2x = x ^4− 4 − 2x such that g(x) = −1/ 2 (x ^4− 2x − 4 ) Will the root be found? (iii) What is the smallest value k that will guarantee convergence if we add −kx to both sides of the equatio

a spherical ice of initial volume 1000 cm^3 is melting at a rate 2 times the surface area. when does it completely melt? volume of the sphere of radius is r=4/3 pir^3 and surface area = 4pir^2 This is a ODE problems Thanks!

Suppose ? is a finite-dimensional with dim ? > 1 and ? ∈ ℒ(?). Prove that {?(?)|? ∈ ?[?]} ≠ ℒ(?).

The surface of a spherical conductor of radius a is kept at a temperature of u(φ)=300K+50K cos(φ). The temperature inside is governed by the Laplace equation. Find an expression for the temperature everywhere inside the sphere. Evaluate the temperature at the center of the sphere.

according to the US census bureau, the population of the US seniors65 and older, in the year 2004 was approximately 36,300,000 people. In the year 2010, it was 40,267,984 people. the senior population was growing at an approximately constant rate during this period.

(a)use this information to express the US senior population as a function of time since the year 2000.

(b) what is slope of your function? what does this mean in the context?

(c) what would this model indicate that the US population of seniors was in the year 2000? ( The actual population in that year was approximately 34,991,753)

(d) what does this model predict the US population of seniors to be for the 2020 census? Do you think this will be an overestimate or underestimate and why?

(e) when does your model predict the US population of senior to be 45,500,000 people?

The probability destiny function is where statistics and probability come together. While there are several different kinds of discrete probability functions (or PDF's), three in particular are most commonly used. These are the binomial, Poisson and hypergeometric. What are the characteristics of each? Where and how are they used? Have you ever seen or even used any of these?

(5) In each part below, find a basis for R4 that contains the given set, or explain why that is not possible: (a) {(1,1,0,0),(1,0,1,0),(0,1,1,0)} (b) {(1,−1,0,0),(1,0,−1,0),(0,1,−1,0)} (c) {(1,1,0,0),(1,−1,0,0),(0,1,−1,0),(0,0,1,−1)} (d) {(1,1,1,1),(1,2,3,4),(1,4,9,16),(1,8,27,64),(1,16,81,256)}

Suppose f : N→N satisestherecurrencerelation f(n + 1) (f(n) 2 if f(n)iseven 3f(n)+ 1 if f(n)isodd . Notethatwiththeinitialcondition f(0) 1,thevaluesofthefunction are: f(1) 4, f(2) 2, f(3) 1, f(4) 4, and so on, the images cyclingthroughthosethreenumbers. Thus f isNOTinjective(andalso certainlynotsurjective). Mightitbeunderotherinitialconditions?3 (a) If f satisestheinitialcondition f(0) 5,is f injective? Explain whyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage. (b) If f satisestheinitialcondition f(0) 3,is f injective? Explain whyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage. (c) If f satisestheinitialcondition f(0) 27,thenitturnsoutthat f(105) 10 and no two numbers less than 105 have the same image. Could f beinjective? Explain. (d) Prove that no matter what initial condition you choose, the functioncannotbesurjective.