Given two sets S and T, the direct product of S and T is the set of ordered pairs S × T = {(s, t)|s ∈ S, t ∈ T}.Let V, W be two vector spaces over F.

(a) Prove that V × W is a vector space over F under componentwise addition and scalar multiplication (i.e. if (v1, w1),(v2, w2) ∈ V × W, then (v1, w1) + (v2, w2) = (v1+w1, v2+w2) and a(v, w) = (av, aw) for any (v, w) ∈ V ×W, a ∈ F).

(b) If dim V = n and dim W = m, prove that dim V × W = n + m by constructing a basis.

1.     Describe your personal view on algebra being a requirement for your degree program. How do you feel about being forced to take math?

2.     Mathematics is part of the liberal arts curriculum, as well as the curriculum for any STEM field. Describe what you think of when you hear the phrase “liberal arts.” What about “STEM?”

3.     Describe what you think the purpose of higher education is.

When Gustavo and Serrana bought their home, they had a 5.1% loan with monthly payments of $870.60 for 30 years. After making 78 monthly payments, they plan to refinance for an amount that includes an additional $35,000 to remodel their kitchen. They can refinance at 4.5% compounded monthly for 25 years with refinancing costs of $625 included with the amount refinanced.

A) Find the amount refinanced. (Round your answer to the nearest cent.)

(b) Find their new monthly payment. (Round your answer to the nearest cent.)

(c) How long will it take to pay off this new loan if they pay $1200 each month? (Round your answer up to the next whole number.)
payments

Type in only your numerical answer to the following problem; do not type any words or letters with your answer. The population of Neverland was 2.3 billion, one hundred years ago. Currently the population is 3.2 billion. What will this population be 100 years from now? NOTE: Round to the nearest tenth of a billion.

The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $170; the quantity demanded each month is 750 when the unit price is $150. The suppliers will market 750espresso makers if the unit price is $110. At a unit price of $130, they are willing to market 2250 units. Both the demand and supply equations are known to be linear.

(a) Find the demand equation.
p =

(b) Find the supply equation.
p =

(c) Find the equilibrium quantity and the equilibrium price.

A group of retailers will buy 120 televisions from a wholesaler if the price is $375 and 160 if the price is $325. The wholesaler is willing to supply 88 if the price is $320 and 168 if the price is $410. Assuming the resulting supply and demand functions are linear, find the equilibrium point for the market. Find (q,p).

The daily demand for ice cream cones at a price of $1.20 per cone is 50 cones. At a price of $2.20 per cone, the demand is 30 cones. Use linear interpolation to estimate the demand at a price of $1.50 per cone.

The table shows the estimated number of E. coli bacteria in a lab dish t minutes after the start of an experiment.

A. Using t as the independent variable, find the model that best fits the data. Round values to the nearest thousandths.

B. How long does it take the population of E. coli to triple?

We can approximate the continuous-time tank model of the previous problem by a discrete model as follows.

Assume that we only observe the tank contents each minute (time is now discrete). During each minute, 20 liters (or 10% of each tank’s contents) are transferred to the other tank.

Let x1(t) and x2(t) be the amounts of salt in each tank at time t. We then have:

x1(t + 1) = 9 /10 x1(t) + 1 /10 x2(t)

x2(t + 1) = 1 /10 x1(t) + 9 /10 x2(t)

Formulate the problem in the form x(t + 1) = Ax(t) where A is a 2 × 2 matrix, then solve for the amount of salt in each tank as a function of time using the eigenvalues and eigenvectors of A.

Sketch the graphs of the amount of salt in each tank as functions of time.

How does your solution compare to the continuous time model?

The walls and ceiling in your bedroom need to be painted, and the painters’ estimates to do the work are far too expensive. You decide that you will paint the bedroom yourself. Below is the information to help you solve the problem:
• The bedroom is 17 ft., 3 in. long by 18 ft. wide, and the ceiling is 9 ft. high.
• The color of paint you have selected for the walls covers 84 square feet per gallon and costs $31.50 per gallon.
• The inside of the bedroom door is to be painted the same color as the walls.
• The ceiling will be painted with a bright white ceiling paint that costs $27.50 per gallon but only covers 73 sq. ft. per gallon.
• Two coats of paint will be applied to all painted surfaces.
• The room has one window, measuring 3 ft., 3 in. by 4 ft., which will not be painted.

1. Because different paint lots of the same color may appear slightly different in color, when painting a room, you should buy all of your paint at one time and intermix the paint from at least two different cans so that the walls will all be exactly the same color. Because all ending values are given in feet, first find the room dimensions in feet that make a good model for this situation.

ANSWERS
LENGTH ft.
WIDTH ft.
HEIGHT ft.

Explain your answer here:




2. Using the measurements found above, label the sides in feet here:













                                                      
3. Using the formula concepts and dimensions from above, find the bedroom’s total painted surface area around all of the walls, including both coats. Do not forget to subtract the window’s area. Also, double the paint to account for two coats.

Show all step-by-step calculations, including the units of measurement, and round your final answers to the nearest whole measurement unit:

ANSWER
Total painted wall surface area

Explain your answer here:




4. Using the formula concepts and dimensions from above, find the ceiling’s total painted surface area, including both coats.

Show all step-by-step calculations, including the units of measurement, and round your final answers to the nearest whole measurement unit:

ANSWER
Total painted ceiling surface area

Explain your answer here:






5. Describe and discuss the strategy, steps, formulas, and procedures for how you will use Polya’s problem-solving techniques to determine how much it will cost to paint this bedroom with two coats of paint (on all walls and the ceiling).

Explain your answer here:






6. Find, individually and as a total, how much it will cost to paint this bedroom with two coats of paint (on all walls and the ceiling).

Show all step-by-step calculations, including the units of measurement, and round your final answers to the nearest whole dollar amount:

ANSWERS
Total cost painted wall surface area
Total cost ceiling surface area
Overall total cost of paint

Explain your answer here:





7. Assuming you can paint 100 sq. ft. per hour, what will be the work time needed to paint your bedroom?

Show all step-by-step calculations, including the units of measurement, and round your final answers to the nearest whole hour amount:

ANSWERS
Total painting time

Explain your answer here:

This week we study complex numbers which include an "imaginary" part. This is an unfortunate name,  because imaginary numbers can be proven to exist and they are very useful for describing certain physical phenomena. Search for one interesting fact about imaginary numbers. This can be their history, application, etc. Be sure to read through your classmates' postings first, duplicate facts will not count.

a. What is your fact?

b. In your secondary responses to your classmates' facts, endeavor to expand our knowledge or understanding of how they affect the world as we know it.

Consider the set of vectors S = {(1, 0, 1),(1, 1, 0),(0, 1, 1)}.

(a) Does the set S span R3?

(b) If possible, write the vector (3, 1, 2) as a linear combination of the vectors in S. If not possible, explain why.

find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse with the given equation. remember to complete the square in oder to accuartely graph the ellipse: 9x^2+6y^2-36x+12y=12