The following is an excerpt from a current work of mine. Writing it in the form of an matrix is merely a very convenient way to see the map concretely.

### Course notes

But it obfuscates the fact that this map is, well, a map, not an array of numbers. To any typical high school student, a matrix is an array of numbers. Then you go into linear algebra in college, and you do more magic with these arrays of numbers.

Then you define the characteristic polynomial as. And then you have the Cayley-Hamilton Theorem in all its black magic: is the zero map. This is terrible. Who gives a — about? Only if you know that the matrices are linear maps does this make sense: is just rewritten with a different choice of basis. In my eyes, this mess is evil. Linear algebra is the study of linear mapsbut it is taught as the study of arrays of numbersand no one knows what these numbers mean. And for a good reason: the numbers are meaningless.

They are a highly convenient way of encoding the matrix, but they are not the main objects of study, any more than the dates of events are the main objects of study in history.

When I took Math 55a as a freshman at Harvard, I got the exact opposite treatment: we did all of linear algebra without writing down a single matrix. During all this time I was quite confused. But I would just like to say for the record that, if I had to pick between the high school approach and the 55a approach, I would pick 55a in a heartbeat.

These will actually follow as corollaries from the complete decomposition of the finite regular representation. In what follows is an algebraically closed field, is a finite group, and the characteristic of does not divide.

Recall that is the vector space of functions from towith addition being defined canonically.Harvard apparently has this notoriously difficult math course that the really advanced frosh students take. I only first heard about it by reading this article recently, excerpt below:. Later that first night, the first problem set is released online: 13 questions, each consisting of multiple sections to make a total of 47 parts.

He figured wrong. As class attendance steadily thins, the workload does not. The first few problem sets each take about 40 hours to complete. The work burden is reason enough for many extraordinarily gifted students to drop.

Case in point: Ameya A. But I realized it was not going to work out. The article analogizes it to be like a fraternity. Of math geeks. But freshman year I took We used a slim, blue, and terse book. The slim, blue, and terse book. And from my perspective, that book had been forged in the depths of hell. Everything in class made sense.

Well, everything in the first five minutes. We had weekly quizzes. I think I usually got a 0, 1, 2, or 3 points on them.

Out of But I endeavored. The belief that I could do it? And at my school, I thought it was rare that anyone went to talk with the Great Professors and ask for help.

### 17 Insane Facts About "Math 55," the Hardest Math Class at Harvard

I was a freshman. The final was hard — I think only 4 or 5 questions — and I left depressed. But I nailed it. Many of my students go through that struggle and frustration in high school. But I can identify with their frustration. It just came later for me. A forum talking about the course.Math 55 is a two-semester long first-year undergraduate mathematics course at Harvard Universityfounded by Lynn Loomis and Shlomo Sternberg. The Harvard University Department of Mathematics describes Math 55 as "probably the most difficult undergraduate math class in the country.

As ofstudents may choose to enroll in either Math 25 or Math 55 but are advised to "shop" both courses and have five weeks to decide on one. A take-home final ends the class. InMath 55 covered almost four years worth of department coursework in two semesters, and subsequently, it drew only the most diligent of undergraduates. Of the 75 students who enrolled in the offering, by course end, only 20 remained due to the advanced nature of the material and time-constraints under which students were given to work.

Kominers, familiar with the stated attrition rates for the course, decided to keep an informal log of his journey through the section: " Throughthe instructor had broad latitude in choosing the content of the course. Infor example, students studied the differential geometry of Banach manifolds in the second semester of Math These topics typically culminated in the proof of the generalized Stokes' theoremthough, time permitting, other relevant topics e.

Loomis and Sternberg's textbook Advanced Calculus[10] an abstract treatment of calculus in the setting of normed vector spaces and on differentiable manifoldswas tailored to the authors' Math 55 syllabus and served for many years as an assigned text. From onwards, the scope of the course along with that of Math 25 was changed to more strictly cover the contents of four semester-long courses in two semesters: Math 25a linear algebra and Math group theory in Math 55a; and Math 25b calculus, real analysis and Math complex analysis in Math 55b.

Fluency in formulating and writing mathematical proofs is listed as a course prerequisite for Math 55, while such experience is considered "helpful" but not required for Math Typical students of Math 25 have also had previous exposure to proof writing through mathematical contests or university-level mathematics courses.

Problem sets are expected to take from 24 to 60 hours per week to complete, [1] although some claim that it is closer to 20 hours. Demographics of students taking this course over the years have been used to study the causes of gender and race differences in the fields of mathematics and technology.

## GED Math Practice Test

Math 55, along with several other high-level mathematics courses, was brought up by Dr. Spencer Reid in a episode of Criminal Minds entitled "Mr. In the eighth episode of the seventh season of the television series "The Doctor," the protagonist Warren references Math 55 when describing the intelligence of his friend Erik.

Warren recounts how, "out of the forty students in the class, Erik was the only one who didn't get 'chunked'. From Wikipedia, the free encyclopedia. Undergraduate mathematics course. Archived from the original on October 19, Retrieved December 9, Harvard Independent. Archived from the original on January 21, October 20, The Harvard Crimson.

Math 55a: Honors Abstract Algebra. December 6, Ury, Dec 6, The Crimson The tribe has spoken.Prerequisites: Mathematical maturity appropriate to a sophomore math class.

Credit Option: Students will receive no credit for 55 after taking Computer Science Introduction to graphs, elementary number theory, combinatorics, algebraic structures, discrete probability theory. Week 1: Propositional logic, quantifiers, rules of inference Sections 1. Week 3: Sets, functions, countability and uncountable sets Sections 2. Week 5: Euclidean algorithm, modular exponentiation, solving congruences, Chinese Remainder Theorem Sections 4.

Week 6: Induction and recursion, recursive algorithms Sections 5. Week 7: Counting, pigeon hole principle, permutations and combinations, binomial coefficients, distributions, generalized permutations and combinations Sections 6. Week 8: Discrete probability theory, conditional probability, independence, random variables Sections 7.

Week 9: Bayes' Theorem and applications Section 7. Week Recurrence relations Sections 8. Week Relations, transitive closure, equivalence relations, set partitions, partial orders Sections 9. Week Optional topics: Cryptography Section 4. Note: Optional topics may be incorporated at appropriate points in the semester, at the instructor's choice. Jump to Navigation. Introduction to mathematical arguments by Michael Hutchings.

University of California, Berkeley. CalNet Login. Search this site:.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.

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.The GED is a secondary option for individuals who did not complete high school and receive their high school diploma.

The GED, recognized in all 50 states, serves as a high school equivalency credential. With this credential, graduates can apply for colleges and entry-level employment positions. You are eligible to take the GED math test if you meet the following requirements: You are at least 16 years old You are not enrolled in high school You have not graduated from high school You meet all of your state's additional requirements ex. Students are encouraged to check out the GED website for their state's prices.

The mathematical reasoning section of the GED test consists of two types of problems, quantitative problem-solving and algebraic problem-solving. The GED math test has two sections. On the first section, which consists of 5 questions, you are not allowed to use a calculator.

For the second section, which includes 41 questions, you may use a calculator. Please note that you must bring your own TIXS calculator. Some questions are fill in the blank or multiple select, where you must select multiple answers, which means multiple points.

Generally, you can take the GED 3 times, and after the 3rd attempt, you have to wait 60 days. Similar rules apply to students taking the test in different languages. For example, if a student takes a test 3 times in Spanish, after the 3rd attempt, there is a day waiting period to take it in English.

**Solving one of the toughest Indian exam questions**

Some states have individual retake rules. For example, in the District of Columbia D. Students are encouraged to check out their state's retesting policy. To pass the GED math test, you must earn a score of at least The highest score isand anything below is considered failing. The GED math test consists of 46 questions.

Some of the many question types include multiple-choice, drag-and-drop, hot spot, and fill-in-the-blank. The rules and procedures for test day vary with each testing center.

Below you will find some common tips to help you on test day. Does the thought of taking the GED math test make you nervous? This comprehensive study guide not only covers the major topics on the GED math test, like variables and line plotting, but it also includes practice test questions.

Making the study experience fun and effective, the Mometrix GED Flash Cards are ideal for quizzing and studying on the go! These all-in-one cards cover all the major topics on all four sections of the GED test. You can also look forward to learning effective test-taking strategies that will make a big difference on test day!

We congratulate you on taking the first few steps toward earning your GED. We look forward to helping you ace the test! All of our resources in one place for one low price!

What type of math is on the GED Test?The 20v max is no slouch but I could always use more power and runtime. Such tools require little redesign and development, as they can be based off the same motors and gearing used in the new budget impact driver.

The new rotary hammer has been confirmed as being listed as brushless in some Dewalt catalogs, but they have yet to comment on that matter. Dewalt probably wants to keep the brushed motor compact drill kits at lower prices so that users with higher budgets and more demanding runtime needs look more towards the brushless kits with 2. The premium drills are now kitted with 4. Chris saysAug 10, 2013 at 11:09 amWell all the big drill kits with the 3.

Double the run time of non brushless.

I will always pay the extra cash for the brushless vs non if given the choice. Bring on the brushless. Milwaukee and Makita (among other brands) have been using brushless technology for years. From what I can see physically, high torque brushless motors have bigger field and armature. I believe dewalt will come out with one eventually after it finds the best balance for brushless size motor and battery use.

One thing that I like about dewalt is that it likes to wait to learn from other technology errors to come out with theirsat least that is what I believe. Of course the heavy duty drill was shown to be more powerful than the compact model.

The company then used testimonials in their marketing. That brushless motor has permanent magnets and it looks to be like your typical canned motor. Makes me wonder if its a longevity issue. Over on electrician talk there have been quite a few M18 fuel drills burning up chucks. People seem to think that the motor puts out more torque than the chuck can handle.

Chris saysAug 11, 2013 at 10:18 amJust ordered myself a 985 bare tool on ebay for 100 bucks Needed a big drill to do some bigger jobs around the house. Thing is I only have 4 of the smaller 1. Power draw is typically proportional to loading.

If it proves to be a hassle to go back and forth to the charger you can always pick up a 3. I have that drill and both compact 1. The drill works fine with the compact batteries. I learned that the hard way and the 3. It is scheduled to be released in October but these types of things can always change.

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