The title of each problem should guide anyone perfectly. However, this is a contradiction with the fact that F is a field since either a or v j need to be zero but neither are. This results in the following and thus the subset is closed under addition. Thus the subset is closed under addition. So we are left with finding a set where scalar multiplication is not closed.

So we should be able to use this to our advantage. Because we have that each of the sets in the intersection is a subspace of Vthen they all contain the zero vector, and therefore, so does the intersection. Now let x be in the intersection of subspaces, which in turn means that x is in each of the individual subspaces and hence so is ax for any scalar awhich of course then means that the intersection contains ax as well, and thus we have the closure of scalar multiplication on the intersection.

Also let y be a vector in the intersection. Thus the sum is also in the intersection, and the intersection is therefore closed under addition. We can similarly arrive at the converse relationship to prove equivalency, and therefore the commutativity of addition of subspaces.

The proof for associativity is virtually the same as the previous proof for commutativity, but simply replacing the commutativity of vector addition with its property of association. Similarly, any subspace of U will be an identity for Ubut note that U will not be an identity for the subspace, outside of the trivial subspace of U itself.

So we can see that the identity is not actually unique. This is due to the fact that each addend contains the zero vector, and thus the sum of two subspaces will always have at least all the elements from the larger of the two or more subspaces. However, as a result of what was previously discussed, there could potentially be many inverses of a subspace since there are multiple identities for a given subspace.

If we were to choose one of the trivial identities, U itself, then any subspace of U would be the inverse of U. This is, of course, unless the subspace is U. So I began to think of previous problems above and specifically thought about how adding a vector space and one of its subspaces would yield the first vector space, and then constructed the following counter-example from knowing that. Futhermore, since it is not possible for polynomials of the form in If I knew of such a theorem, that is what I would have used.

Find The Bijection In this proof, we let aband c all be elements of S A or S A 0context will define which one, and notation such as a k will be an element of A which belongs to the sequence a at the k th position. Prove S A is a ring. These, slightly annoying, but necessary manipulations follow first for commutativity, then associativity.

There are a few key things to notice here. The first is a general thing about changing indicies. Nonetheless this property is only true for summations when the sum is a commutative operation, and since we are in the world of a ring, Awe are free to switch around indicies!

So we take advantage of this in the changes from equation The other thing to note is that the distributive law of A allowed us to go from equation Giving to us that the distributive law holds for S Aand thereby satisfying our final property to prove that S A is a ring. Prove that S A 0 is a ring. Here, like subspaces, we just need to check that S A 0 contains both sum and product identities and is closed under both of the sum and product operations. Since the sum identity is all zeros, and the multiplicative identity is a one followed by all zeros, then they are both contained in S A 0.

Find an isomorphism from A to S A 0.Please join StudyMode to read the full document. On the other hand, highly specific requirements could turn auditing into mechanical evidence gathering, void of professional judgment.

From the point of view of both the profession and the users of auditing services, there is probably a greater harm from defining authoritative guidelines too specifically than too broadly. Rossi and Montgomery's primary ethical consideration is their professional competence to perform all of the audit work for filing with the SEC.

The filing with the SEC, in addition to normal audited financial statements, will require completion and registration with the SEC of Form S- 1 which includes an audited summary of operations for the last five fiscal years as well as many additional schedules and descriptions of the business.

Each quarter subsequent to the filing, Form Q must be filed; and within 90 days of the end of each fiscal year Form K must be filed with the SEC. In addition, Form 8-K must be filed whenever significant events have occurred which are of interest to public investors Thus the events Ac and Bc are mutually exclusive.

Thus, there is nothing that is not a part of Ac or Bc. Hence, Ac and Bc are collectively exhaustive. Thus, Ac and Bc are not collectively exhaustive. TCO 1 What is the purpose behind the five primary activities in the value chain? First, info flow diagrams and flowcharts argon the two most much utilize development and musical accompaniment tools employ today. Second, since systems developme nt is super complex, DFDs and flowcharts ar! Second, since systems development is extremely complex, DFDs and flowcharts are tools that are used to create order from chaos and complexity.

Points The flow diagram and the flowcharts are the two more common systems use. Also when it comes to the system development they can get very complex meaning that the DFD and the flowchart are tools that can be used to create order form chaos and complexity.

Second, since systems developme nt is super complex, DFDs and Express each of these quantifications in English. There exists a student in school, who has visited North Dakota. All students in the school have visited North Dakota c. Not all students in the school have visited North Dakota. The case between Beauty and Stylish involves concept of a valid contract, pre-contractual statements, express term and misrepresentation.

A valid contract is established between Beauty and Stylish when an offer is accepted and there is intention for both parties to create legal relations.

An offer refers to the expression of willingness of the offerer to be contractually bound by an agreement if his or her offer is properly accepted.Yum-Tong Siu siu math. Office Hours. MWF 1 p. Walter Rudin, Principles of Mathematical Analysis. Springer, 2nd Edition.

Course Assistant. Le ile fas. Prerequisites and Comparison with Math Math 55 is intended for students with significant experience with and enthusiasm for abstract mathematics. Its syllabus is similar to that of Math Math 55 differs from Math 25 not so much in the choice of topics as in the level of exposition.

The Mathematics Department offers these courses at separate hours so that you can "shop" both, which you are strongly encouraged. You may switch between Math 55 and Math 25 without penalty for the first three weeks of the semester. From our past experience, in nearly all cases it is best to resist such a temptation. Weekly problem sets will be given on Friday and due in class the following Friday. Late homework will not be accepted. You are encouraged to discuss the course with other students, your Course Assistant, and me.

It is much easier to learn mathematics if you have other people who will help you test your understanding and overcome problems. It is fine to discuss homework problems with other students, but you should always write your homework solutions out yourself in your own words and understand them. There will be two in-class quizzes that will test your recollection of basic concepts. Each quizz will count for the equivalent of one homework problem set.

The Final Exam will be a take-home exam. For the final take-home exam you will be on your honor to work completely on your own. Two-third of the course grade will be based on the homework problem sets and the quizzes. The final take-home exam will account for almost all of the remaining one-third of the course grade, with class participation used mostly to decide on borderline cases. The course is not graded on a curve. The grade is based only on the performance of each individual student and not on the relative standing of the student in the whole class.

The assignment of grades is not constrained by any rule of a fixed percentage for any particular grade. Mathematics 55a Syllabus. Springer, 2nd Edition Course Assistant.Many files on this site in PDF format. You will need to download a free copy of the Acrobat Reader to read them. Click the following icon to obtain a free copy of the Acrobat Reader.

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If this happens, carefully select the appropriate version of the reader.

## 55 Lesson 6 Homework

Assignments and activities will be accumulated on this page throughout the semester. Please return often as this page will be updated frequently. This is a link to the Errata Page for the textbook. If you see any errors in the text not listed on this page, send me an email description, including page number. Notebook for Part 1. Notebook for Part 2. Notebook for Part 3. Notebook for Part 4. Notebook for Part 5. Notebook for Part 6.

Notebook for Part 7. Notebook for Part 8. Notebook for Part 9. Notebook for Part Visit Hands-on Start to Mathematica for a beautiful introduction to people just beginning to use Mathematica. In Differential Equations with Boundary Value Problemsperform the following tasks: Answer the questions posed in exercises 1, 2, and 4 on pageusing Mathematica to draw the required sketches.

Obtain a printout of the Notebook and attach it to your homework. Use the Text command to mark each equilbirum solutions with its equation. On pagesdraw the sketches requested in exercises 8, 10, 12, 14, 15, 18, 19, and 22 by hand.Problem Solution. Find all eigenvalues and eigenvectors of T. Hence all eigenvectors of T must be of the form x, … ,x. Since the dimension of the image of T is kthen there is no linearly independent set of vectors of size larger than k in the image.

Since eigenvectors for a given eigenvalue have the form then all vectors in a given set of eigenvectors corresponding to distinct eigenvalues will be in the image of Tand thus limited in size to kwhich in turn limits the number of such eigenvectors to k.

In this case we have the following. The proof of the reverse direction is equivalent to the above, making the obvious substitution. Prove that ST and TS have the same eignevalues. Prove that T is just a scalar multiple of the identity operator. Hence, we can see that every eigenvalue of each vector in the above basis is the same. Assume by way of contradiction that V is not W. We know that W is a subspace of V since both the image and the kernel of P are subspaces of Vwhich implies, by our assumption, that V is not a subspace of W.

**Math 55 intro**

Thus there exists at least one basis vector of V that is not in W. Hence our assumption was incorrect and V is indeed W.

## David Arnold

Furthermore, since the eigenvalues are distinct, then the corresponding eigenvectors will be linearly independent, which in turn implies that the eigenvectors will also be a basis since the vector space in question has dimension two. Hence our eigenvectors are Saving you the eye-sore of a calulation that is finding the inverse of Prentice Hall.

Upper Saddle River NJ: Linear Algebra Done Right 2nd Ed. New York NY: Ideas about a prelude on the class takes between 24 to 6th ed. Ann levine, 95, q - math 55 is long first-year undergraduate mathematics - download as well, you now! Harvard's math.

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Hi chris, mathematics course at harvard extension school let the question that you but today i heard a.The course is one many students dread, while some sign up out of pure curiosity, to see what all the fuss is about. Reportedly, on the first day of each semester, the class is standing-room only.

But a select few consider it a rite of passage that their mathematical mettle. The two-semester-long-course—which is made up of "Honors Abstract Algebra" Math 55ain the fall, and "Honors Real and Complex Analysis" Math 55bin the spring—is far tougher than its unimposing name might have you believe. But, by all accounts, it's totally worth going through the ordeal. Since very few of us will ever get a chance to start the course, let alone finish it, here are 17 insane things you probably don't know about the craziest academic experience most of us could ever possibly imagine.

He passed. You could take Math 21, which is taught by graduate students and where homework usually takes a reasonable three to six hours per week.

Or, you could opt to take Math 55—which gives you 10 times the amount of homework.

### 55 Lesson 6 Homework

So, basically, it's a full-time job, plus a part-time job—and that's on top of the rest of your course load, internships, any actual part-time gigs, and requisite Harvard partying. University of Pennsylvania professor David Harbatera fellow former 55er, adds: "It's probably safe to say there has never been a class for beginning college students that was that intense and that advanced.

You can expect to see students bailing from Math 55 on a regular basis. The class size shrinks to half its original size or less before the semester is over. According to one student who took Math 55 inand kept a running tally of attendance, "We had 51 students the first day, 31 students the second day, 24 for the next four days, 23 for two more weeks, and then 21 for the rest of the first semester after the fifth Monday.

By all accounts, those who make it through the first five weeks—the end of Harvard's add-drop period—become thick as thieves. They have to; the coursework demands it. So they hang out together, they party together, they do homework together.

Homework happens in a fluorescent-lit common room in Thayer Hall, a dormitory on the north side of Harvard Yard. Steve Ballmere. No attitude of 'We're better than everyone else. People who have been through it don't really forget it….

It's like changing fraternity rings. It identifies you. Most Math 55 students produce to page problem sets each week. They have to produce so much, that students are sometimes discouraged from showing too much of their work. As one former 55er explains"Lots of students write out the solutions in lemma-theorem form, proving everything from rock bottom. I did this also.

This makes your problem set enormous. This is not so easy to do in an undergraduate proof class, where nearly all the proofs are of obvious facts. In Williams' bookhe describes Richard Stallman's Math 55 ending the semester with 20 students, eight of whom would go on to become future mathematics professors.

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