Most people did well with the definitions. That's great! The True/False questions were a bit of a killer for some people. Part A revealed a lot of weaknesses that were not revealed by the other questions. The Final will target some of the areas that people need to work on, so will contain variations on the following midterm questions:
Important dates: There will be a midterm on Friday November 6. I am teaching 3 sections of Math 308 this quarter and the finals take place at different times and places. You should check the dates. I believe the times and places are: 308G 8:30 Thursday, December 17; 308H 2:30 Monday, December 14; 308J 2:30 Tuesday, December 15. The final will be two hours long. You must use a blue book for the midterm and final.
Office Hours: Monday 9:00-10:00 and Friday 9:00-11:00 in Padelford C-418, and by appointment.
I often find office hours the most enjoyable part of the course. I get a chance to know you, you get a chance to know me. My job is to help you learn this material. You are often likely to feel lost and perhaps stupid. That is normal. I often feel lost and stupid when I am working in a new area of math. It is part of the process. But I can provide you with guidance and help. If I can stretch some medical metaphors (my wife is a physician), it is no good coming to see the doctor or start taking the medicine once the illness is terminal. Catch it in the early stages, at the first sign of a sneeze or ache, and the chances for recovery are good. (I am not as fearsome as I might appear---remember that I was once your age, feeling small on a big campus, and that I have experienced and continue to experience the struggles we all have as part of life. And, heck, the math struggles are among the smaller ones! So, why not start praticing in the small arena of math to say "I need help".)
Other linear algebra resources.
Here is Terry Tao's linear algebra course page. I recommend you look at his notes.
Here are some multiple choice questions on a range of topics, some of which are to do with linear algebra, that you can use to test your understanding.
Definitions. I give you some advice about writing definitions. I always ask for a lot of definitions in my exams because they are very important. If you don't know the definitions you will surely fail this course. Look at this document early in the quarter.
Set notation and language. We will use the basic language of sets and functions in this course so I have written some notes about what you need to know. You can take an online test to see how well you understand the concepts.
My notes. I have typed up some notes for the course. They follow the lectures pretty closely. There are some differences from the text book. I hope they help. If you find typos, have quesions, or would like me to explain a little more about some topic please let me know. Thanks!
Pop Quizzes. There will be unannounced 10-15 minute quizzes every 2 or 3 weeks (depending on my energy level and free time). Here are some examples.
Grading. Your grade will be based on the homework, the midterm, and the final. Your best six homework scores will contribute 25%, the midterm will contribute 30%, and the final will contribute 45%.
I have posted two old midterms, here, and here , and an extra long practice midterm. The postings also include answers and comments on the questions, answers, and common errors. The very best advice I can give you is to print these off at the beginning of the quarter and study them assiduously. Especially, look at the questions that relate to the different topics as we encounter them week by week. Check that you can do the questions. Check that you understand my answers and comments. Ask questions if you don't understand. You ignore this advice at your peril.
I want you to know what I want you to know. I want you to be able to answer the questions on these midterms. At least 75% of the questions on your midterm will be the same as those on these old midterms.
Final. Here is an old practice final. It is far longer than the real one will be, but you will get a sense of your abilities by trying it.
Homework is due at the start of class each Monday. Except in extraordinary circumstances, late homework will not be accepted. If you miss the homework deadline, that homework will be one of those which does not contribute to your best six.
I expect all homework to be neat and legible. Part of the homework is to make your solutions easy to understand. Write neatly. Space out your work. If you have more than one page for your solutions staple them together. Unstapled homework solutions will not be accepted. The 10 points you get for just doing all the problems will also involve whether your homework is neat and tidy and legible. So, even if you do all problems but it looks a mess or is difficult to understand you will be awarded less than the full 10 points. For example, if the paper you hand in is tattered down the left-hand side you wil receive less than 10 points. Part of your job is to make the grader's life easy by handing in work that is legible and easy to understand.
You should check the grader's arithmetic. If you think there is an error in the grading return the homework to me with a written explanation of your complaint. I will return it to the grader. If that does not resolve the problem, then they can come to you.
What follows is a list of Homework exercises for the course, and the due dates. I have kept the comments made by the grader for a previous Math 308 course I taught. They might be helpful.
The grader's comments: Problems 51 and 8 were the killer ones for many people. There were lots of computational errors on other problems as well. One of the main reasons why students had lots of errors is because they perform too many operations at a time using the same row. Several students have problems understanding the ``consistency condition'' (see page 6 for the definition, page 19 for a discussion of how to recognize an inconsistent system, and section 1.3 for more details). There were a few who gave answers but no work; they got zeros.
Based on their performance on this homework, I think quite a few students need to improve their elementary row operation techniques. There are a lot of errors made on row reducing a matrix to its echelon form. On # 23, and 24, I was surprised to see the number of students that had problems reducing the augmented matrix. Some students are confused with the definition of a non-trivial solution, and whether a system is consistent or inconsistent. On # 33, a lot of the students couldn't solve the equation correctly, # mainly due to the difficulty of reducing the augmented matrix.
The grader's comments: Be careful when copying down the statement of each homework exercise. Write grammatically and mathematically correct statements. Just using equal signs would be a step forward. Use nice paper and staple the pages together. Some people would do better if they broke each problem into smaller steps. Most students know that in #27, A is not equal to B, because cancelation doesn't work on Matrix, and there's no division in Matrix, but some of them didn't give a counter example. On #28, a lot of the students have trouble telling the flaw: "the argument assumes that if the product of two matrices is zero, then one of the factors must be zero". On #46, many students don't know the purpose of this problem is using Theorem 10, that (A+B)^T= A^T +B^T.
Almost everyone did #52 from 1.5 correctly. For #44 sec 1.6, most students had trouble step (c). They either forgot to mention about using property 2, addition is associative or didn't substitution A+C by B+C. #46 is the worst problem. A lot of students didn't say (x-y)T= xT- yT, and xTy= yTx. Overall, the performance on this homework is better than last weeks.
The grader's comments: For 1.7 #23, some students are confused about the definition of non-singular. they did not reduce the matrix (D+F) to the echelon form, and didn't point out that x1=x2=x3=0. and that it has a unique solution. For 1.7 # 51, I saw a lot of different solutions. Most of the students did it the way different from the solution manual. When I graded, I looked at if the students show that they understand what is linear independence. If the student did not explain what makes {v1,v2,v3} a linearly independent set, I took off points. I also looked at whether they proved how {v1,v1+v2 v1+v2_v3} has the same solution as {v1,v2,v3}, which is when a1=a2=a3=0. Some of them used one example of a matrix that can be reduced to identity matrix. I gave them 4/5. If the answer is too vague or lack of explanation, I gave them 2 or 3 points. For 1.8 #6. Almost all the students had no problem setting up the system of equations. Most errors were made during the process of elementary row operations. For 1.8# 28, some students don't know how to set-up the matrix to solve for a, b, c and d. For 1.9 #54, most students did well because this problem is more straight-forward. A few students have no clue how to solve this and just left it blank.
Additional problems: At the end of each chapter are some nice additional problems. Here are some at the end of Chapter 1 that I found attractive/interesting. Supplementary Exercises: 4,5,9,10,11,13,18. Conceptual Exercises: 1,2,3,4,5,8,9,13,14. These are not assigned as homework problems. However, your goal in this course should be to attain some mastery of the material. Your grade will reflect the degree to which you have achieved such mastery. Trying these problems is an excellent way of working towards such mastery. If you find these problems difficult there is probably some key point that you are not yet understanding. It is better to discover that in the privacy of your own home than when sitting an exam. Once you do understand that key point you will find all subsequent problems easier.
The grader's comments: For 3.2 #28, some students didn't recognize that property c2 is not satisfied because if a<0, then ax is not in W, which illustrates that (a4) is also not satisfied. One or two students also said that m1-m3 are not satisfied because the scalar can be negative. I think m1-m3 are satisfied since the properties didn't say anything about R^n. For 3.3, problem #42 most students found the matrix A, but didn't conclude that W is a subspace of R^3. I gave them points when they use Theorem 5 " if A is an (m*n) matrix and if R(A) is the range of A, then R(A) is a subspace of R^m". For #51, some students don't understand how to interpret the intersection of N(A) and N(B): it means that x must be in both N(A) and N(B). But most students did well on this problem. ,p> For those who did the homework completely, I'd say they did a good job. They seem to understand the material well. On 3.2 # 28, property c2 and a4 are not satisfied because x_2>=0, and by multiplying a coeffient a<0 will give us a negative x_2. some students mention that the m properties are also not satisfied, that's not true. the m properties do not deal with R^n. so those are fine. On 3.3 #58. most students did well. those who made errors on the problems either had the negative sign forgot or copied the wrong numbers from the problems. In the future, they should be more careful with the numbers to avoid losing unnecessary points.
The grader's comments: For 3.4 #24, part a) some students had problems reducing the matrix formed by the vectors in S. They also need to point out that x_4 is an unconstrained variable. so the basis is {S1,S2,S3}. part b) some students forgot to take the transpose of the final reduced form. For #37, a), most students know that any set of 3 or more vectors in R^2 is linearly dependent, and is not a basis for R^2. b) some students don't know how to obtain a contradiction by using the "Hint"- {e_1, e_2} is a linearly independent set. if a basis has only one vector, then e_1 and e_2 are dependent. For 3.5#18, by far, this one has the highest correct rate! Almost no one had trouble with this one. Everyone did a good job, except a few students didn't show any work.
For 3.4 #24, most students did well. part a), it's important that students show x_4 is the independent variable, the desired basis is {u_1,u_2,u_3}. I took off 2 pts for some students who didn't do part b) For 3.5 # 37, I didn't see many students did it the way the solution manual does. For most students, I took off 2 points if they didn't say rank(A)= rank([AIb]), and use Theorem 11 to claim that the system Ax=b is consistent. Since I saw so many students did it in different ways, you might need to make some clarifications on this one.
The grader's comments: For 3.6#16, students know pretty well about using the Gram-Schmidt process to generate orthogonal set. The few mistakes are due to mis-calculation. 3.7#18, most students had e_1 is an orthonormal basis for W. Some didn't know how to use equation 4 of example 3: T(v)= (v^T e_1)e_1= [a, 0, ]T. #28 , i don't think anyone had problem figure out A. rank=3, nullity=0.
3.6#25, need to compare ||x+y||^2 and (||x||+||y||)^2. some students expanded (||x||+||Y||)^2 wrong. 3.7#21, I gave only 1-2 points if students don't show the work on how they obtained T(x). I saw other ways of doing it besides the solution manual method, might need some clarification on this one.