Linear Algebra Spring 2012
Comments on the first exam

Scores for the first exam have been posted on blackboard.  Exams will be returned in class on Monday, 2/20/12.  It is important that you come to class to pick up your exam because you are assigned to write up a set of corrections that will be due Monday, 2/27.  Exam corrections are a required component for the course portfolio.  Specific directions are posted here.  

When you get your exam, please double check that I have added up the points correctly and also that the score on your paper is the same as the one posted on blackboard.  I try hard not to make errors totaling or recording scores.  But if an error has been made, I will want to correct it as soon as possible.

The exam scores are shown in the histogram below.

exam 1 histogram

The histogram shows that the class as a whole did pretty well.  Nearly 1/2 of the scores are at 90 or above, and almost 2/3 of the scores are at 87 or above.  At this point everyone is earning a satisfactory grade, and almost everyone is on track for a final grade of B or better. 

Anyone who is unhappy with his or her performance should have a discussion with me.  I may be able to suggest ways to study more effectively. 


Comments on Theoretical Aspects of the Course and the Exam

Students frequently ask why I ask them to state definitions and theorems on exams.  "Are we supposed to memorize all of that material?" they often want to know.  Ideally your knowledge of the material should be learned more meaningfully than by rote memorization.  But not all memorization is bad or pointless.  Sometimes it is the most effective way to make a factual foundation for a subject a part of your knowledge.  But however you do it, it IS important that you learn the basic facts.  The comments below explain in greater detail why this factual foundation is important to your mastery of linear algebra, or any mathematical subject.

Emphasis on Definitions and Theorems.  Many people missed points on items referring to statements of definitions and theorems.  This material is important, and unfortunately, is not adequately emphasized in many math classes.  Ultimately, the point of the class is to learn the concepts, and in particular, what is true.  The facts of the course are of two types. Some are conventions that we agree to adopt.  These are the definitions.  They are the basic assumptions about concepts, as well as technically defined terminology and notation. The rest of the facts are forced on us by logic.  There is no freedom or controversy about these.  Given the basic assumptions, the statements given in theorems are inescapable consequences. 

Theorems vs Definitions.  It is worth knowing which are which, since basic assumptions can be revised, but theorems are inescapable once you have the basic assumptions.  So, for example, if we are unhappy about facts that emerge from a given set of definitions of independence and spanning, we cannot change those facts directly.  We have to go back to the definitions and change those.  Also, the definitions cannot surprise you.  They are a matter of convention and we simply accept that they are useful.  But theorems can be surprising, unexpected, can give us powerful tools, and sometimes are neither obvious nor easy to prove.  By the same token, because definitions are not based on logic, you cannot figure them out on the basis of what you think might be true.  So there really is no substitute for learning exactly what the definitions state.  But once you do know the definitions, you can work out many of the theorems by logic.  For all of these reasons, it is important to know what are definitions and what are theorems.

Proper Mathematical Definitions vs Definitions in Common Usage.  In everyday language, when you are asked to define something, that means you should give an essay explaining what something means.  You might include ancillary facts and/or examples.  In mathematics, definitions have a much more restricted meaning.  They say concisely and unambiguously what the defined concept is.  They do not include examples.  They do not include a motivating rationale.  They d o not include additional facts that we can prove using the definition.  They just say what a specific term or notation or concept is.  And in particular, a definition of a concept must be specific enough that it allows you to determine in any specific instance whether or not the concept is present.  For example, the definition of linear independence must allow you to determine with clarity whether a particular set of vectors is or is not linearly independent.

Correct Statements.  It is also important to know the statements of definitions and theorems correctly.  That doesn't necessarily mean word for word memorization.  But it does mean that you have to know the meaning in great detail, and then to convey that meaning correctly.  And that also requires correct and careful use of language.  Linear algebra students often find this difficult.  They mix up the terminology that concerns matrices, systems of equations, and sets of vectors.  Examples of common errors are referring to
If you use any of these, you are mixing up meanings and using language imprecisely.  You may know what you mean, but you are not communicating that knowledge accurately.   And when you hear a term used in lecture or see it in the reading, if you are confused about exactly what it means, you will not fully understand whatever point is being made.   Say for example that you are not quite clear on what is a parametric vector form for the solution set to a system of equations.  If I say in lecture that for a homogeneous equation the parametric vector form for the solution demonstrates something or other, you will not be in a position to see how or why my statement is true.

The statements of theorems and definitions in the book are carefully formulated to say precisely the correct thing, and to say it in a concise way.  If you really understand what those statements mean, and why they are true, then you have gone a long way toward mastering this subject.

Mathematics is Like the Law.  This distinction between everyday language and the language used in mathematics may be easier to understand in terms of an analogy.  In legal matters, the ultimate authority in deciding whether something is permitted or not is the actual language of the statutes that have been enacted (plus the case law provided by past legal decisions).  But we also have common interpretations of what the law means.  We all know, for example, that elected officials are not supposed to accept payments in exchange for making decisions in a certain way.  These types of descriptions help us understand what the law means, but they are not acceptable in a legal proceeding.  For a court case, you have to go back to the exact language of the law (the definitions) or to decided case law (the theorems).  Where you or I might think it is obvious that an official has broken the law based on press reports, that doesn't necessarily translate into a conviction in court.  In a legal proceeding the lawyers have to show that an infraction has occurred relative to the actual wording of the law or the accepted precedents.  In the same way, in mathematics, it may help us to understand linear independence if we have a concept saying no vector in the set is a linear combination of the other vectors in the set.  But when it comes to establishing some property of independence, we have to go back to the letter of the law -- the definition.  And just as a legal expert has to know what the law (and accepted precedents) actually says, so it is important for a linear algebra expert (that is, you) to know what the definitions (and the theorems) actually say. 

Proper Use of Math Language is Important.  Some students find this insistence on correct use of language annoying.  Some think that I am just way too picky a grader.  I understand that point of view, but there is a method in my madness.  The fact is, extremely careful use of language is a significant contributor to the amazing record of mathematics in avoiding error and identifying truth.  In just about any other field you can name, theories come and go, and what was accepted lore one day becomes discredited or at least reinterpretted the next.  Not so in mathematics.  Once we know what is true (once we have proved theorems), those results persist pretty much forever.  But obsessive care with the precise use of language is one of the prices we pay for that.  As a math teacher, I want my students to at least understand in overview what the mathematical method is and how it works.  You should know not only what tools linear algebra gives you, but how we know that these tools are valid, and when they are applicable.  Part of this requires careful use of language and terminology.  I know that this kind of attention to detail is not appealing to everyone.  But if you want to really understand math, that is what is required.

So, when you study for an exam, go back through the book and review all statements of definitions and theorems.  Make sure you understand what they say, and that you are able to restate them accurately.  You should also be able to give concrete examples that illustrate the meanings of definitions and theorems, as well as non-examples.  For instance, for the definition of linear independence, you should be able to give examples both of sets that ARE independent, and sets that are NOT, and why.  For the theorem that says a transformation is one to one if and only if the columns of the matrix are independent, you should be able to give an example of a one to one transformation and show that the columns of its matrix are independent, and you should also be able to give an example of a transformation that is NOT one to one and show that its matrix has columns are NOT independent.