. Let T : R n → R m be a linear transformation and A the standard matrix of T. (a) Let BN = {~v1, . . . , ~vr} be a basis for ker(T) (i.e. Null(A)). We extend BN to a basis for R n and denote it by B = {~v1, . . . , ~vr, ~ur+1, . . . , ~un}. Show the set BR = {T( r~u +1), . . . , T( ~un)} is a basis for range(T) (i.e. col(A)). Conclude that dim(col(A)) = n − dim(Null(A)). (b) Show that row(A) = Null(A) ⊥ (i.e. ker(T) ⊥). (c) We have established that for any subspace W of R n (or of any finite dimensional vector space V ), dim(W) + dim(W⊥) = dim(R n ) = n (or dim(W)+dim(W⊥) = dim(V )). Conclude that dim(col(A)) = dim(row(A)).

​Kroger, the​ country's leading​ grocery-only chain, added a line of private label organic and natural foods call Simple Truth to its stores. If​ you've priced organic​ foods, you know they are more expensive. For​ example, a dozen conventionally farmed​ Grade-A eggs at Kroger costs consumers$1.81​,whereas Simple Truth eggs are priced at $4.1 per dozen. One study found​ that, overall, the average price of organic foods is 85 percent more than that of conventional foods.​ However, if prices get too​ high, consumers will not purchase the organic options. One element of sustainability is organic​ farming, which costs much more than conventional​ farming, and those higher costs are passed on to consumers. Suppose that a conventional egg​ farmer's average fixed costs per year for​ conventionally-farmed eggs are​ $1 million per​ year, but an organic egg​ farmer's fixed costs are five times that amount. Further assume that the organic​ farmer's variable costs of​$2.22.2per dozen are twice as much as conventional​ farmer's variable costs.

Most large egg farmers sell eggs directly to retailers. Using​ Kroger's prices, what is the​ farmer's price per dozen to the retailer for conventional and organic eggs if​ Kroger's margin is

25 percent based on its retail​ price?

The price the farmer sells a dozen of conventional eggs is ​$_______________

Recall that in class we defined, for any set A, P(A) := {B : B ⊆ A}, which we called the power set of A.

a) (10) Show that, if a set A has n elements (where n ∈ {0, 1, 2, 3, 4, . . .}), P(A) has exactly 2 n elements.

b) (10) Use #4(any infinite set S has a countably infinite subset) to show that, if A is an infinite set, P(A) is uncountable.

Show that the set of all n × n real symmetric matrices with zero diagonal entries is a subspace
of R^n×n

Prove (for example, by calculating each side)

(a) Q[ √ 2, cube root(2)] : Q = [Q[ √ 2] : Q] · [Q[ cube root(2)] : Q]

(b) Q[ √ 2, fourth root (2)] : Q not equal to [Q[ √ 2] : Q] · [Q[ fourth root (2)] : Q]

Post a 1- to 2-paragraph analysis of a graph from the news media. Your posting should include the following: A graph from the news media that you think may be closely modeled by a linear, quadratic, polynomial, rational, logarithmic, or exponential function If you are using an electronic resource, either provide the link or scan and upload it. Make sure you scan as one of the following file types: *.bmp, *.gif, *.jpg, *.jpeg, and *.png. An explanation of which type of function the graph most closely models A detailed description of the behavior of your chosen graph, such as increasing and decreasing intervals, max or min values, intercepts, domain and range, end behavior, symmetry, even or odd behavior, the type of equation, asymptotes, and restricted values Does your graph go through the origin? Does it have an x-intercept and/or a y-intercept? Why or why not? At what x-values does the graph rise or fall? Is there a reason behind the behavior of the graph? More detail is better.

1. Use backward substitution to solve:

x=8 (mod 11)

x=3 (mod 19)

2. Fine the subgroup of Z24 (the operation is addition) generates by the element 20.

3. Find the order of the element 5 in (z/7z)

Molson currently sells 41 different brands of beer in Canada. Labatt currently sells 17 different brands. The manager at Mike's Place needs to choose 5 Molson brands and 5 Labatt brands to sell. How many options do they have? The 10 beers (5 Labatt, 5 Molson) selected in the previous question must be placed in a line on a display shelf so that no two Molson products are adjacent and no two Labatt products are adjacent. How many ways are there to do this? Continuing from the previous question, suppose two of the brands selected by the manager were (Labatt) 50 and (Molson) Export. The display shelf can not have a bottle of 50 adjacent to a bottle of Export. How many ways are there to do this while still avoiding two adjacent Molson products and two adjacent Labatt products? How many of the arrangements from the previous question have the bottle of 50 and the bottle of Export among the first (leftmost) 5 bottles?

Direct product of groups: Let (G, ∗_G) and (H, ∗_H) be groups, with identity elements e_G and e_H, respectively. Let g be any element of G, and h any element of H. (a) Show that the set G × H has a natural group structure under the operation (∗_G, ∗_H). What is the identity element of G × H with this structure? What is the inverse of the element (g, h) ∈ G × H? (b) Show that the map i_G : G → G × H given by i_G(g) = (g, e_H) is a group homomorphism. Is it injective? Surjective? Do the same for the map i_H : H → G × H given by i_H(h) = (e_G, h). (c) Show that the map π_G : G × H → G given by π_G (g, h) = g is a group homomorphism. Is it injective? Surjective? Do the same for the map π_H : G × H → H given by π_H (g, h) = h. (d) Prove that the image of i_G is the kernel of π_H, and that the image of i_H is the kernel of π_G

The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population P₀ has doubled in 5 years.
Suppose it is known that the population is 9,000 after 3 years.

What was the initial population P₀? (Round your answer to one decimal place.)

P₀ = ____________________

What will be the population in 10 years? (Round your answer to the nearest person.)

_________________persons

How fast is the population growing at t = 10? (Round your answer to the nearest person.)
________________ persons/year

solve the following LP. Formulate and algebraically solve the problem. Show all steps.

what is the new optimal z value

max z=65x1+35x2+20x3

8x1+6x2+x3<=48

4x1+2x2+1.5x3<=20

2x1+1x2+0.5x3<=8

x2<=5

x1,x2,x3>=0

interpret the meaning of the shadow prices

Writing Prompt(s)

One method for solving a system of first order linear differential equation such as

x ′ = a x + b y y ′ = c x + d y

is to take the derivative of the first equation and use the second equation to ``decouple'' the system and create a second order equation, which we can solve using our previous techniques. Does this always work? If not, what conditions on the constants a, b, c, and d must be enforced? If it does work, we can then arrive at an equation for x(t). How do we proceed in finding an equation for y(t)?

2. Obi-wanandAnakinarechasingZamWessel.Obi-wanandAnakinareatthepoint(325,675,561) while Zam is at the point (765, 675, 599).

   1. (a) Find parametric equations for the line of sight between the Jedi and the bounty hunter.

   2. (b) If we divide the line of sight into five equal segments, the heights of the buildings at the four intermediate points from Obi-wan and Anakin to Zam Wessel are 549, 566, 586, and 589. Ignoring other buildings, can Zam see Obi-wan and Anakin?

Suppose there are two lakes located on a stream. Clean water flows into the first lake, then the water from the first lake flows into the second lake, and then water from the second lake flows further downstream. The in and out flow from each lake is 500 liters per hour. The first lake contains 100 thousand liters of water and the second lake contains 200 thousand liters of water. A truck with 500 kg of toxic substance crashes into the first lake. Assume that the water is being continually mixed perfectly by the stream.

a) Find the concentration of toxic substance as a function of time in both lakes.

b) When will the concentration in the first lake be below 0.001 kg per liter?

c) When will the concentration in the second lake be maximal?

Determine the root of

f (x) = 10.5X² - 1.5X - 5

by using Newton Raphson method with x0 = 0 and perform the iterations until ɛa < 1.00%.

Compute ɛt for each approximation if given the true root is x = 0.7652.