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Pavel Grinfeld - Hello Again, Linear Algebra: A Second Look At The Subject Through A Collection Of Exercises And Solutions

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Hello Again, Linear Algebra: A Second Look At The Subject Through A Collection Of Exercises And Solutions
Author:
Pavel Grinfeld
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Insightful Texts
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Year:
2014
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Thank you for your interest in my book. This book is supported by a forum at http://bit.ly/HALA_FORUM_AM. Some of the topics are illustrated in the videos found at http://bit.ly/HALA_PLAYLIST_AM.
This book is for current linear algebra students looking to master the concepts of the subject, and for those who have taken it in the past looking for a refresher. Its an easy read which aims to illustrate concepts with examples and exercises.
Linear algebra is a grand subject. Because it is fundamentally different from any high school mathematics, and because of the wildly varying quality of instructors, not all students enjoy learning it. In either case, whether you loved it or hated it, it takes several passes to learn linear algebra to the point that it becomes one of your favorite tools, one of your favorite ways of thinking about practical problems.
This little textbook invites you on your second pass at linear algebra. Of course, your second pass may take place alongside your first pass. You may find this textbook particularly useful if you are studying for a test. Our goal is to take a step back from the mechanics of the subject with an eye towards gaining a larger view. A larger view, however, is achieved in small steps. We are not hoping for a big revelation but for a few small aha! moments. It simply takes time to put together the grand puzzle of linear algebra. You will get there, and the point, as the cliche goes, is to enjoy the ride.
Psychology in mathematics is everything. I chose the topics according to the impact I feel they would make on your relationship with linear algebra. The textbooks utmost goal is to make you feel positively about the subject. You will find that some topics are surprisingly simple, others surprisingly tough. Some topics have important applications, others have none at all. Some were well presented in your linear algebra course, others skipped altogether. However, I hope you will find that all topics bring you a little closer to the subject of linear algebra.

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Chapter 1 Introduction

Linear algebra is a grand subject. Because it is fundamentally different from any high school mathematics, and because of the wildly varying quality of instructors, not all students enjoy learning it. If you hated it, I blame the instructor. There I said it. In either case, whether you loved it or hated it, it takes several passes to learn linear algebra to the point that it becomes one of your favorite tools, one of your favorite ways of thinking about practical problems.

This little textbook invites you on your second pass at linear algebra. Of course, your second pass may take place alongside your first pass. You may find this textbook particularly useful if you are studying for a test. Our goal is to take a step back from the mechanics of the subject with an eye towards gaining a larger view. A larger view, however, is achieved in small steps. We are not hoping for a big revelation but for a few small aha! moments. It simply takes time to put together the grand puzzle of linear algebra. You will get there, and the point, as the cliche goes, is to enjoy the ride.

Psychology in mathematics is everything. I chose the topics according to the impact I feel they would make on your relationship with linear algebra. The textbook's utmost goal is to make you feel positively about the subject. You will find that some topics are surprisingly simple, others surprisingly tough. Some topics have important applications, others have none at all. Some were well presented in your linear algebra course, others skipped altogether. However, I hope you will find that all topics bring you a little closer to the subject of linear algebra.

There are several points that this textbook emphasizes over and over again. Some of these points are made explicitly. Others are more difficult to put into words and it is my hope that these points become clear as a result of considering the many examples analyzed here. Therefore, I would like to make these points right here with minimal explanations.

1. The applications of linear algebra are extraordinarily broad. The subject applies to any type of objects that can be added together and multiplied by numbers. These objects are united by the term vector and include pointed segments, polygons or more generally functions, sets of numbers organized in a column, matrices, currents, voltages, substances in chemical reactions, strains and stresses in elastic bodies, sound signals, portfolios of financial securities. Need I go on?

2. Linear algebra studies the commonalities among these objects. However, focusing on the commonalities is counterproductive if one ignores the differences. It is that the fact that these objects are all uniquely different that makes linear algebra such a remarkable tool. We therefore invite the reader to accept and treat each object on its own terms. If the problem relates to pointed segments which we call geometric vectors, by all means let us have a geometric discussion and talk about straight lines, lengths, and angles. Do not rush to associate a geometric vector in the plane with a pair of numbers. It is not necessary and, in fact, detrimental to clarity, to do so. Unfortunately, just about every textbook in the land does this and in doing so obscures much of the psychology of linear algebra. Similarly, when dealing with

, lets talk about entries, rather than lengths and angles. Sets of numbers have entries and nothing else. Sets of numbers don't have lengths and directions, and when they do eventually acquire these notions, it doesn't happen in the corrupted way presented in most textbooks. Finally, functions are functions, and polynomials are polynomials. There's no need to think of polynomials as "vectors" of coefficients. Polynomials are cool on their own terms. Other than adding them and multiplying them by scalars, you can differentiate them and even dilate them. Can you do that with "vectors" of coefficients?

3. Let the problem dictate the basis, or say no to Cartesian coordinates! This point is related to the preceding point and applies to more than the subject of linear algebra. There seems to be a lamentable tradition to refer the space of our everyday physical experience to Cartesian coordinates, whether or not it is the best coordinate system and, even more lamentably, whether or not coordinates are needed in the first place. If I may speak for Descartes, he would have been appalled by this state of affairs! His ingenious invention of coordinates was meant to unite the worlds of algebra and geometry. The knee-jerk reaction to translate a physical problem to numbers, typically by introducing a Cartesian coordinate systems, and often doing so tacitly, usually leads to loss of simplicity, vision, clarity, and understanding.

In linear algebra, choosing a coordinate system is equivalent to choosing a basis and the Cartesian syndrome manifests itself by choosing the orthonormal basis Hello Again Linear Algebra A Second Look At The Subject Through A Collection Of Exercises And Solutions - image 2

Hello Again Linear Algebra A Second Look At The Subject Through A Collection Of Exercises And Solutions - image 2

in the plane:

Hello Again Linear Algebra A Second Look At The Subject Through A Collection Of Exercises And Solutions - image 3

or the basis Hello Again Linear Algebra A Second Look At The Subject Through A Collection Of Exercises And Solutions - image 4

for the space of cubic polynomials:

Hello Again Linear Algebra A Second Look At The Subject Through A Collection Of Exercises And Solutions - image 5

or the basis in First of all as was just discussed in point 2 dont resort to a - photo 6

First of all as was just discussed in point 2 dont resort to a basis - photo 7

First of all as was just discussed in point 2 dont resort to a basis in the - photo 8

First of all, as was just discussed in point 2., don't resort to a basis in the first place, thereby converting a "real-life" problem to numbers, unless the reasons for doing so are clear to you. Good reasons for doing so, for instance, are described in Chapter With Or Without a Basis. Secondly, when the time is right to convert the problem to

by decomposing all elements of the problem with respect to a basis, let the problem dictate your choice of basis.

This is much more than a matter of practicality. If a basis is chosen poorly or, worse yet, introduced unnecessarily, the artifacts of the basis will obscure the analysis of the problem. My first linear algebra professor Bill Thurston states in his famous essay On Proof And Progress In Mathematics that the value understanding is greater than that of the formal correctness of a solution. For an engineer (even if you are not an engineer, I'll think of you as an engineer: you want to create things, don't you?) the value of understanding is even greater because the products of your labor are required to work. So if don't wish for the bridge you built to collapse second guess yourself when you instinctively reach for a basis when presented with a problem. Heal yourself of the Cartesian syndrome. Here! I included a portrait of Descartes, because I think that he would agree with me.

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