Let T1 be the reflection about the line 4x?3y=04x?3y=0 in the euclidean plane. What is the standard matrix A of T1 ?
What are the two eigenvalues and corresponding eigenspaces of A ?
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(3) Let V be a vector space over a field F. Suppose that a ? F, v ? V and av = 0. Prove that a = 0 or v = 0.
(4) Prove that for any field F, F is a vector space over F.
(5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1, a2, a3 ? R} of polynomials of degree ? 3 is a vector space over R (with respect to the usual addition and scalar multiplication of polynomials).
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(2) Let Z/nZ be the set of n elements {0, 1, 2, . . . , n ? 1} with addition and multiplication modulo n. (a) Which element of Z/5Z is the additive identity? Which element is the multiplicative identity? For each nonzero element of Z/5Z, write out its multiplicative inverse. (b) Prove that Z/nZ is a field if and only if n is a prime number. [Hint: first work out why it’s not a field when n isn’t prime. Try some small examples, e.g. n = 4, n = 6.]
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Prove Desargues’ Theorem for the case where AC is parallel with A′C′ on the extended Euclidean plane.
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Find answer for the following floor and ceiling functions.
⌊66.6⌋ = ?
⌈66.6⌉ = ?
⌊-66.6⌋ = ?
⌈-66.6⌉ = ?
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Decide which formula to use and than solve
A. Sara borrows $1200 at 8 % simple interest. If the loan is for 5 months, what is the total amount he pays back (This is sometimes called the maturity value)?
B.Sara later decides to deposit $9000 at 7.5% per year compounded annually, and would like to know how much she will have after 10 years.
C. Sara's husband wants to invest $1000 at the end of each quarter at 9% compounded quarterly, and would like to know (1) how much will he have after 10 years? (2) How much interest will he earn after 10 years?
D. Sara is considering depositing $600 at the end of each semi-annual period, for 5 years earning interest of 8%. She would like to know how large a one-time lump sum deposit she could make, at the same rate, to have the same amount of money after 5 years.
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Determining the Selling Price of a Home Determining how much to pay for a home is one of the more difficult decisions that must be made when purchasing a home. There are many factors that play a role in a home’s value. Location, size, number of bedrooms, number of bathrooms, lot size, and building materials are just a few. Fortunately, the website Zillow.com has developed its own formula for predicting the selling price of a home. This information is a great tool for predicting the actual sale price. For example, the data to the right show the “zestimate”—the selling price of a home as predicted by the folks at Zillow and the actual selling price of the home for homes in Oak Park, Illinois. The graph below, called a scatter diagram, shows the points (291.5, 268), (320, 305), . . . , (368, 385) in a Cartesian plane. From the graph, it appears that the data follow a linear relation. 1. Imagine drawing a line through the data that appears to fit the data well. Do you believe the slope of the line would be positive, negative, or close to zero? Why? 2. Pick two points from the scatter diagram. Treat the zestimate as the value of x and treat the sale price as the corresponding value of y. Find the equation of the line through the two points you selected. 3. Interpret the slope of the line. 4. Use your equation to predict the selling price of a home whose zestimate is $335,000. 5. Do you believe it would be a good idea to use the equation you found in part 2 if the zestimate is $950,000? Why or why not? 6. Choose a location in which you would like to live. Go to www.zillow.com and randomly select at least ten homes that have recently sold. (a) Draw a scatter diagram of your data. (b) Select two points from the scatter diagram and find the equation of the line through the points. (c) Interpret the slope. (d) Find a home from the Zillow website that interests you under the “Make Me Move” option for which a zestimate is available. Use your equation to predict the sale price based on the zestimate. can i get the full answer of this question please
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Prove that n, nth roots of unity form a group under multiplication.
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7. Describe how to achieve the best score by Smith-Waterman in linear space.
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Use the formula for computing future value using compound interest to determine the value of an account at the end of 5 years if a principal amount of $5,000 is deposited in an account at an annual interest rate of 3% and the interest is compounded monthly.
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2. New uses of polynomials are being discovered frequently. In fact, Tamás Erdélyi, a Texas A & M professor made significant contributions as recently as 1995.
a. Find another mathematician who advanced the study of polynomials and share what you learned. Be sure to read through your classmates' postings first, duplicate facts will not count.
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For the following exercises, suppose that x2 = 25 + 36 − 60 cos(52) represents the relationship of three sides of a triangle and the cosine of an angle.
Draw the triangle.
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Explain what is meant by deductive reasoning. Give an example of the Law of Detachment and the law of Syllogism.
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1 Use Cramer’s rule to determine the solution of the system of equations:
4x1 + 3x2 + 2x3 = 8
−x1 + 2x3 = 12
3x1 + 2x2 + x3 = 3
2. Consider the following system of equations involving a parameter k:
(2 − k)x1 + kx2 = 4
kx1 + (3 − k)x2 = 3
(a) For what value(s) of the parameter k does the system have a unique solution?
(b) Solve the system of equations using Cramer’s rule.
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