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Paper Assignment for Algebra Study Questions
Paper Assignment – Algebra Study Questions
Mathematics 160.102 Algebra (Auckland, Manawatu, and Distance) MASSEY UNIVERSITY School of Fundamental Sciences and School of Natural and Computational Sciences
Due at 11 p.m., Assignment 2 Semester One, Tuesday, May 26, 2021
1. Given that z = 1 + 2i is a root of p(z) = z 4 5z 3 + 13z 2 19z + 10, determine all of p(zroots )’s and use them to represent p(z) as a product of (a) four linear factors and (b) real linear and real irreducible quadratic components. Use the command roots to check your roots in Matlab.
2. Plot the four roots of z 4 + z 2 + 1 in the complex plane using the subsitution w = z 2.
3. With n = 3, use de Moivre’s Theorem (ei) n = ein, or (cos+isin) n = cos(n)+isin(n), to get the triple angle formulae that represent sin(3), cos(3), and sin(3), respectively, in terms of sin() and cos().
4. Compute the determinant of 1 1 1 0 0 2 1 0 1 2 2 1 1 1 1 4 by using simple row operations (row echelon form).
In Matlab, verify your answer.
5. Prove that if is an eigenvalue of A, then 2 is an eigenvalue of A2 for all square matrices A.
6. Prove that if is an eigenvalue of A, then 1/ is an eigenvalue of A1 for all invertible square matrices A.
7. When Matlab is asked to supply the eigenvalues of a n n matrix (such as this one), what does it do?
1 1 0 1 ) that has a smaller number of linearly independent eigenvectors than n? When a huge n n matrix has fewer than n linearly independent eigenvectors, how could you use Matlab to figure out? In Matlab, demonstrate your method.
8. The Matlab program randn(n) generates a nn matrix with each entry representing a random number from the normal distribution.
(a) Make a 4 4 matrix and find the eigenvalues and eigenvectors with Matlab.
Check in Matlab that the eigenvectors matrix may be used to diagonalize the matrix. How many eigenvalues are there in the real world? Do you always receive the same amount of real eigenvalues when you repeat the experiment with various random numbers?
(b) Make a big such matrix (without printing it; for example, use A = randn(100); where the semicolon prevents the result from being printed) and plot its eigenvalues as points in the complex plane. What do you think you’ve noticed? What happens if you change the value of n?
9. Even on a computer, computing the eigenvalues and eigenvectors for huge matrices the way we’ve been doing it by hand is too expensive. The power approach, which is faster, is as follows:
Step 1: Pick a nonzero x0 starting vector.
Step 2: For k = 0, 1, 2,…, let xk+1 = Axk
Step 3: For k = 0, 1, 2,…, let bk = x> k xk+1 x> k xk.
The sequence b0, b1, b2,… tends to the eigenvalue of A with the biggest modulus, and xk tends to an eigenvector. Let A = 3 1 1 12 0 5 4 2 1 and x0 = 1 1 1
Compute b0, b1, b2, b3, and b4 in Matlab.
How do they stack up against A’s biggest eigenvalue?
Paper Assignment for Algebra Study Questions
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Excellent Quality 95-100%
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Introduction
45-41 points The background and significance of the problem and a clear statement of the research purpose is provided. The search history is mentioned. |
Literature Support 91-84 points The background and significance of the problem and a clear statement of the research purpose is provided. The search history is mentioned. |
Methodology 58-53 points Content is well-organized with headings for each slide and bulleted lists to group related material as needed. Use of font, color, graphics, effects, etc. to enhance readability and presentation content is excellent. Length requirements of 10 slides/pages or less is met. |
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Average Score 50-85% |
40-38 points More depth/detail for the background and significance is needed, or the research detail is not clear. No search history information is provided. |
83-76 points Review of relevant theoretical literature is evident, but there is little integration of studies into concepts related to problem. Review is partially focused and organized. Supporting and opposing research are included. Summary of information presented is included. Conclusion may not contain a biblical integration. |
52-49 points Content is somewhat organized, but no structure is apparent. The use of font, color, graphics, effects, etc. is occasionally detracting to the presentation content. Length requirements may not be met. |
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Poor Quality 0-45% |
37-1 points The background and/or significance are missing. No search history information is provided. |
75-1 points Review of relevant theoretical literature is evident, but there is no integration of studies into concepts related to problem. Review is partially focused and organized. Supporting and opposing research are not included in the summary of information presented. Conclusion does not contain a biblical integration. |
48-1 points There is no clear or logical organizational structure. No logical sequence is apparent. The use of font, color, graphics, effects etc. is often detracting to the presentation content. Length requirements may not be met |
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Paper Assignment for Algebra Study Questions