Why you should read this
In what follows I'll explain how I graded the first homework assignment. Similar questions will appear on other homework assignments, and on the exams I will give in the course. I will grade later work in the course in ways similar to what I've done. I believe the standards are reasonable. You may certainly discuss them with me generally and, of course, I can explain any specific applications.

Homework is good. You can practice and be corrected so that when something worth more ("higher stakes") occurs (an exam?) you'll be more likely to do well. More points on homework will help your course grade. And, by the way, students who are serious will at least look at all of the suggested problems, since I will use them as well as the problems I ask to be handed in as models for some exam problems.

Should you do homework? Well, part of me hopes not (for the greedy reason see the discussion in the next section) but most of me (the almost fictional good teacher) hopes that you will. You decide, of course. The decision and consequences are yours.

Random grading (?)
I last taught this course almost 20 years ago. At that time, sections of 250 had between 20 and 25 students. The course was more intimate (!), and interaction between the instructor and students could occur more naturally and be more supportive of learning. Now ... well, now we have a bigger football stadium, etc.: big time Rutgers.

I currently do not have a grader for this course. I assigned 12 problems to be handed in. I decided that was too many for me to grade (currently there are more than 80 students in the two sections I am teaching). I used the Maple random function to "flip a coin" with equally balanced heads (grade the problem) and tails (skip the problem). The result was that I graded 7 problems (am I a winner?): 1.1: 56, 81; 1.2: 15, 33, 49, 62, 69. Each problem was worth 4 points so that the highest possible score was 28. I will not accept late homework. The grading (and summarizing, here) took almost a full working day.

Neatness and professional presentation
People who handed in more than 1 page sometimes just folded over a corner of the papers to hold them together. This is unreliable, so I stapled these pages. In the next assignment, I will penalize students who hand in more than 1 page not permanently fastened. In the assignment after that, I will not read such submissions. Hey: it is time to grow up. Neatness does count in the real world. People don't like professional presentations that waste time. In addition, I will not read problem solutions where the writing is difficult to understand. It is your job to present results carefully and neatly. I will set and follow reasonable standards.

True or false
Several of the problems I graded were "True or False". Let me consider in some detail the grading of two of these problems. The remarks are meant to guide your efforts generally. Such problems will occur on our exams.

• About a statement that is supposed to be true
  I'll discuss problem 56 in section 1.1. The general instruction is " ... determine whether the statements are true or false." Specifically, here is problem 56:

  If A is a matrix for which A+A^T is defined, then A is a square matrix.

  This statement is TRUE. Reporting True with no substantiation will earn only 1 of 4 points. I don't think you can verify the truth of this statement (a general statement!) with an example. You must give reasoning supporting your assertion. Also, asserting and verifying the converse is not the same as verifying the given statement. Writing the definition of a square matrix does not verify the assertion. These actions will not earn credit. You must show the logical connection between "A+A^T is defined" and "A is square". A lengthy discussion is not needed. Here is one solution.

  Suppose A is an m x n matrix. Then A^T is an n x m matrix. Because we are told that A+A^T is defined, the sizes of A and A^T must agree. The number of rows of A is m and the number of rows of A^T is n. Therefore n=m. Since the number of rows of A is the same as the number of columns of A, then A must be a square matrix.

  The logical flow of this paragraph is from "A+A^T is defined" to "A is square". Generally, statements that are TRUE must have some verification or supporting discussion in order to earn full credit.

  There are situations in linear algebra where looking at a collection of special cases is enough to determine the truth of an implication. For example, in digital signal processing, where the scalars are a finite collection of "numbers", it is possible (but frequently neither feasible nor clever) to enumerate and verify all possible instances of an assertion. Generally that is not possible in Math 250.
  

• About a statement that is supposed to be false
  Now I'll discuss problem 62 in section 1.2. Again, this is True or False, and the statement to be analyzed follows.

  If A is an m x n matrix and u and v are vectors in Rⁿ such that Au=Av, then u=v.

  This statement is FALSE. To show that a general assertion is false, it is sufficient to give an example.

  Suppose A=[0] (that's a 1 x 1 matrix!), u=[0], and v=[1]. Then Au and Av are both equal to [0] but u≠v.

  I note that a number of students gave the same example as in the Student Solutions Manual with no citation. Grading of such solutions in the future is discussed below under Plagiarism. Please read that.
   
• How to discover (!) whether a statement is true or false
  Most mathematicians, and, indeed, most people who want to understand complicated situations, begin by analyzing some examples. After learning about "enough" examples, they then can conjecture (guess!) whether or not an assertion is true. If they believe the assertion is true, they will consider the examples they have studied and attempt to see the common features so that a general argument (not involving specifics of numbers) can be given. If an example is found showing that the assertion is false (usually called a counterexample) then that example can be reported along with the statement that the implication is false.

  How fast or easy analysis of examples goes depends on the experience of the people involved. More experience is good, and the way you get experience is ... by working at it. The most successful individuals I know always begin with the simplest situations. Almost surely students copied the example for #62 that they gave from the Student Solutions Manual because it is most unlikely that exactly the same numbers would be reported by those who really were trying to investigate the situation. The example I gave above is silly but very simple. It works, so why not use it?

Explanations
Assertions must be supported by computations or explanations to earn full credit. Ideally explanations should be written in complete English sentences. I realize that English may not be the family language of some students and I have some sympathy. However English is the language of instruction at Rutgers and all students should be able to give short explanations (orally or in writing) in English. I doubt I would be able to pass a similar linear algebra course in Chinese (in fact, I'm sure I wouldn't be able to!). Maybe I could in French. But that's irrelevant: we are at Rutgers, and communication is important. I won't attempt to make sense of weird sentences or fragments of sentences.

Plagiarism/academic dishonesty
Copying on quizzes and exams is cheating. The punishment can be severe. Students should be familiar with the Rutgers Interim Academic Integrity Policy. The situation with regard to homework is perhaps not as clear so it needs some discussion.

I want to encourage students to work together. Therefore I have posted a list of students in sections 1 and 5. But I don't want students to just copy each others' work. That would be plagiarism. Students may work together but should write up solutions to be handed in independently.

A more interesting, more horrible, and, yes, sometimes more amusing situation occurred because of the existence of the Student Solutions Manual. I did not have a copy of that object while I was grading the homework. I now have a copy. Student answers to at least two of the homework problems I graded were strongly influenced by entries in the Solutions Manual. One problem was the true/false question (#62 in 1.2) discussed above. I think many students merely copied what was in the Solutions Manual and submitted that.

A more serious problem was #81 in 1.1, which asked why every 3 x 3 matrix could be written as a sum of a symmetric and a skew-symmetric matrix. That was certainly the most difficult of the 12 problems I assigned and I was not completely happy when my coin-tossing indicated that I should grade the problem. Many students copied the Student Solutions Manual's answer. However, they copied it in a fashion similar to how I would copy, say, a sentence in Russian (another language I don't speak or read -- sorry). Since I don't know Cyrillic letters my copied sentence would have imperfections. Students who copied the complicated answer to #81 frequently left out parts of expressions. They clearly did not know what the heck they were doing! This is horrible, embarrassing (or it should be!), and maybe even humorous. Looking at a few examples of the situation (even for the 2 x 2 case) would have helped. Sending me e-mail (allowable!) might also have been helpful.

In the future, no credit will be earned for copying answers in the Solutions Manual. If you want to use what's there, read the entry enough to understand it, then paraphrase it ("express the same message in different words") using different variables. In the case of an example, invent your own example which can be similar to what's given in the Solutions Manual.

In this course, direct copying of the Solutions Manual, especially without attribution, is indeed plagiarism. See the discussion in the Academic Integrity document about citation and attribution. Would you like to work on a team with people who would report your work or discoveries to "the Boss" as their own? You should always cite sources. Please do this here and elsewhere. You also will probably learn little by copying a solution which weakens a major benefit of doing homework.

About definitions
I am adding this section on 2/1/2011 since I have spent the morning reading the answers to my first quiz which requested a definition. I anticipate that I will ask for definitions on other quizzes and on exams and therefore I'd like to help people cope with requests for definitions.

If I asked a very quiet middle school child "What is a square?" the response might be

Four sides ... equal

On the first quiz I asked for a definition of A^T, the transpose of A. The previous sentence had established that A itself was an m x n matrix whose entry in the i^th row and j^th column was a_ij.

An answer should include one or more complete English sentences. I believe that the answer should begin by describing the type of object that A^T is. So I want to know that A^T is a matrix, and is not a number or a bird or a frog. I think that any answer should include this. Then I need to know something about the size of A^T, and, finally, I need a description of the entries of A^T. So the ingredients are these: matrix, size of the matrix, and entries of the matrix. There should not be any confusion between the matrix itself and its entries, and the size should be carefully stated.

The information required cannot, I believe, be adequately communicated entirely in symbols. Also, you can't "overload" the symbols. A number of students tried to use the equality sign, =, as an abbreviation for the word "is". This is incorrect. Many people gave examples. Examples are rarely complete definitions, and a list of examples is not sufficient here. Examples can be a useful teaching tool, but they are not official definitions in this course.

If you are asked to define a frog, your answer should almost surely begin with the words "A frog is" and go on with complete English sentences describing what you mean by frog. The description should be clear enough so that people can't misunderstand (the word "frog" in English actually has many different meanings, according to the Oxford English Dictionary and you need to be careful!). Mathematics is an example of a strongly typed language: you've got to tell me what kind of object you are describing, and that's an important part of the description.

Here is a growing list of the items defined so far in the course.

Grading the second quiz
This section was written on 2/8/2011 after I graded the second quiz.

First, a comment: the "work" involved in doing the questions on this quiz is logic+arithmetic. I really don't believe that people who got low grades (7 or less) can likely understand what's going on in class, and probably won't be too successful in this course. Many people got 10's (thank you!). If you got a low grade or missed the quiz, consider this a warning notice.

How did I grade? Since my reliability in arithmetic is not high, mostly I looked only at the answers. Part b) was easier, so I graded it first. I looked for a 4 x 4 identity block (1 point) and after that I looked at the four numbers in the last column and took points off when things were wrong.

Part a) was more difficult to grade. There were certainly many people who got "my" answers (actually, Maple's answers, here). But as I wrote on the practice page many answers can be correct. Some I recognized immediately (a sign change, etc.). I saved up the answers I did not recognize (those which were in row echelon form -- answers not in row echelon form received low grades). I then took each of these answers and typed them into the rref command of Maple. Frequently I found that students were correct. Good, correct work deserves credit, and I was happy to give it. I deducted points if the reduced form was far away from the correct answer (that probably meant there were more errors in the elementary row operations). If someone feels very unfairly treated by this method of grading, please let me know. (Hey, it is only a quiz, and we will have many of them.)

Maintained by greenfie@math.rutgers.edu and last modified 1/28/2011.