#### The Quotes đ

I’m so tired but I can’t sleepđ

Standin’ on the edge of something much too deep~ Sarah McLachlan (Lyrics from I Will Remember You)Â

What is the difference between statistics, machine learning, AI, and data mining?đŹ

If there are up to 3 variables, it is statistics.Â

If the problem is NP-complete, it is machine learning.Â

If the problem is PSPACE-complete, it is AI.Â

If you don’t know what is PSPACE-complete, it is data mining.

~ Dan LevinÂ

Essentially, all models are wrong, but some are usefulđ~ George BoxÂ

A handful of observationsâsome erudite, some otherwiseâfrom the trenches of Artificial Intelligence (AI) and Deep Learningđ»

I just got kicked out of Barnes and Noble for moving all their classic statistical theory books to the religious section.Â

Do Neural Networks Dream of Strictly Convex Sheep?Â

They call me Dirichlet because all my potential is latent and awaiting allocationÂ

Batch algorithms: YOLO*, Online algorithms: Keep Updates and Carry On (*You Only Learn Once)Â

~ Courtesy of @ML_Hipster and @bigdatahipster

Make no mistake about it: Computers process numbers, not symbols. We measure our understanding (and control) by the extent to which we can arithmetize an activityđŸ~ Alan J. Perlis

#### Preamble â±

To get the lay of the land for the awesome books reviewed in this essay, let’s dive right into the pic collage below. It shows the top five deep learning *foundational* books currently available on the marketâand not to worry, yet, about what exactly *foundational* means in this context. There will be plenty of time to uncover exactly that in this essay, among many other things đ

#### Introduction â

### The Artist

The other day, somebody called me something that nobody had *ever* called me before: an artist. There are, at least, two dimensions to why that praisefulÂ pronouncement startled meâin a very pleasant way of courseâand I owe you an explanation for what is turning out to be a rather unusual start to an essay on deep learning. Consider, then, the following dimensions of the artistic startlement:

- “Real Artists Ship”: In this memorable phrase from his bestseller entitled
*Linchpin: Are You Indispensable?*(p.101), Seth Godin was reminding us of the importance that shipping a productâbe it software, prose, widget, art, or craftâshould truly occupy in our collective consciousness đą đ đ đ - We programmers and technologists, too, need inspiration from time to time; inspiration isn’t for artists alone. Lest anyone’s jaw dropped at that observationâsaying in response, hey we’re a cerebral and logical bunchâI will to out on a limb and say that as a community we’re at
*least*as much artists as we are a bunch of logical, methodical, and goal-driven individuals. To underscore that precise point, I’ll remind you that there’s an*Art*in Donald Knuth’s legendary and eponymous magnum opus entitled*The Art of Computer Programming*(Addison-Wesley), which is actually a book-set composed of four intense volumes đ đ đ đ

### Awareness

And to bring closure to that artistic startlement, I’ll add that there may *well* be other dimensions that perhaps registered subliminallyâand here I’m reminded of Bollas’ haunting phrase when he mentioned about the “unthought known”âbut of which I’m not quite aware at the moment. Referring here to what Wallin had in mind when he noted how:

In his final book on attachment, Bowlby quotes Freud who remarked on the characteristic response of the patient who has become aware of something “forgotten”: “As a matter of fact I’ve always known it; only I’ve never thought of it” (Bowlby, 1988, p. 101). Perhaps Christopher Bollas (1987) who coined the evocative phrase “the unthought known” was reading the same passage from Freud.Â

~ David J. Wallin PhD (Attachment in Psychotherapy, Guilford Press)

How about all that for the beginnings of an essay on…Â *Deep Learning*?! Wait a second, lest we get ahead of ourselves đ

### Deep Learning

So I have more than a passing interest in deep learning. And casting a glance back at my personal journey, we’ll soon dive deep into five deep learning books that have proved immensely helpful to me in grokking this intriguing field at the *foundational* level. Okay, so what I’ve got in mind when I use the word *foundational* is the following composite thinking, all melded together:Â

In the context of deep learning, afoundationalunderstanding is what’s gained when you have achieved competency and fluency in the use of the mathematical toolbox that powers thefoundationsof deep learning. Basically, what we’re looking at is a conceptual toolbox in which linear algebra is center squareâhelpful techniques for manipulating groups of numbers simultaneously. The awesomeness of linear algebra has its origins in that it provides structures like vectors and matrices to hold numbers, along with powerful rules to add, subtract, multiply, and divide those numbers. Essentially, it equips you with the wherewithal to slice and dice effortlessly through mountains of data in the quest to identify patterns that interest you â đ đ

#### Deep Learning (Foundational) Books: The List đ

With the backgrounder out of the way, let’s move into the essay properâwe’ll soon be taking an opinionated look at the following deep learningÂ *foundational*Â books, in turn:

*Linear Algebra: A Modern Introduction*(Brooks Cole) by David Poole đł*Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation*(Pragmatic Programmers) by Mark C. Chu-Carroll đ*How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics*(Princeton University Press) by William Byers đ*Matrix Analysis and Applied Linear Algebra*(SIAM: Society for Industrial and Applied Mathematics) by Carl D. Meyer đŹ*Thinking Mathematically 2nd Edition*(Pearson) by J. Mason, L. Burton, K. Stacey đĄ

#### 1. *Linear Algebra: A Modern Introduction* (Brooks Cole) by David Poole đł

### A Phenomenally Good Introductory Book

If you carefully peer into the pic above, I want you to know that it’s no accident that the phenomenally good book on deep learning, which was published earlier this yearâperhaps *the* book on the subject, simply entitled *Deep Learning* (The MIT Press) by Goodfellow, Â Bengio, and CourvilleÂ and one that we’ll dive into in a *future* installment in this series of essaysâserenely stands guard in the background, while the nuts-and-bolt *Linear Algebra: A Modern Introduction* (Brooks Cole) by David Poole basks in the limelight with brazen effrontery đŹ

*foundational*. And at the core of that foundationâthe bedrock if you willâis the subject of linear algebra. At this time, I’m not aware of a book that presents that subject better thanÂ

*Linear Algebra: A Modern Introduction*Â đ

### A Learning Experience Imbued With Pleasure

Frankly, I’m not aware of a kinder, gentler, *and* intelligentÂ approach to introducing yourself to linear algebra than through a study ofÂ *Linear Algebra: A Modern Introduction*. This book is all you need to create for yourself a swimmingly good experience, all at your own pace, all on your own time â°

Poole shares the gist of this fine book with clarity and enthusiasm in noting that

I want students to see linear algebra as an exciting subject and to appreciate its tremendous usefulness. At the same time, I want to help them master the basic concepts and techniques of linear algebra that they will need in other courses, both in mathematics and in other disciplines. I also want students to appreciate the interplay of theoretical, applied, and numerical mathematics that pervades the subject.

### Let’s Get Ourselves Acquainted

To acquaint you better with the tremendously valuable foundational aspects ofÂ *Linear Algebra: A Modern Introduction*, let’s have ourselves a peek at its table of contents:

Chapter 1. VectorsChapter 2. Systems of Linear EquationsChapter 3. MatricesChapter 4. Eigenvalues and EigenvectorsChapter 5. OrthogonalityChapter 6. Vector SpacesChapter 7. Distance and ApproximationAppendix AMathematical Notation and Methods of ProofAppendix BMathematical InductionAppendix CComplex NumbersAppendix DPolynomials

Answers to Selected Odd-Numbered Exercises

### The Joy of Learning

Look, I’m a sucker forÂ *adorning*âsome might sayÂ *embellishing*, though I’ll politely beg to differâmy essays with topical quotes,Â images, and excerptsÂ to create a pleasurable reading experience. So I was pleased and right at home with the format I found inÂ *Linear Algebra: A Modern Introduction*. Here is Poole introducing you to the book in the Preface with this delightful and witty quote:

The last thing one knows when writing a book is what to put first đ»

~ Blaise Pascal (PensĂ©es)

*Answers to Selected Odd-Numbered Exercises*, toward the end ofÂ

*Linear Algebra: A Modern Introduction*

Answers are easy. Itâs asking the right questions [thatâs] hard đ

~ Doctor Who “The Face of Evil“, By Chris Boucher BBC, 1977

### Key Definitions and Concepts

Each chapter ofÂ *Linear Algebra: A Modern Introduction*Â ends with a succinct roundup of key definitions and conceptsâhere, for example, are the key definitions and concepts that accompanyÂ *Chapter 4*. Eigenvalues and Eigenvectors đŁ

adjoint of a matrix, 276 algebraic multiplicity of an eigenvalue, 294 characteristic equation, 292 characteristic polynomial, 292 cofactor expansion, 266 Cramerâs Rule, 274â275 determinant, 263â265 diagonalizable matrix, 303 eigenvalue, 254 eigenvector, 254 eigenspace, 256 Fundamental Theorem of Invertible Matrices, 296 geometric multiplicity of an eigenvalue, 294 Gerschgorin disk, 319 Gerschgorinâs Disk Theorem, 321 Laplace Expansion Theorem, 266 power method (and its variants), 311â319 properties of determinants, 269â274 similar matrices, 301

### The Applications of Linear Algebra

Is that good stuff or what, to help you consolidate key concepts? Oh, did I even mention the excellence of the *applications* of linear algebra you’ll find liberally sprinkled across the book? In the words of Poole who is cool and doesn’t drool

I have not been stingy with the applications: There are many more in the book than can be covered in a single course.However, it is important that students see the impressive range of problems to which linear algebra can be applied.I have included some modern material on finite linear algebra and coding theory that is not normally found in an introductory linear algebra text. There are also several impressive real-world applications of linear algebra and one item of historical, if not practical, interest; these applications are presented as self-contained “vignettes” (italics mine).

### Judgment

*and*elevate yourÂ foundationalÂ skills to pursue deep learning like no other book probably probably could. Again, the uncompromising commitment to help you conceptualize exactly whatÂ makes linear algebra tick is evident and writ large throughout the pages ofÂ

*Linear Algebra: A Modern Introduction.Â*In the process of reading it, learning ceases to be a chore and instead becomes true pleasure. Yes, itÂ

*is*Â one of those rare books đ

#### 2. *Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation* (Pragmatic Programmers) by Mark C. Chu-Carroll đ

### Appreciating the Pervasiveness of Math

Deep learning or otherwiseâactually, *especially* for deep learningâit will serve us all well to remain mindful of the pervasiveness of math throughout the fabric of technology, science, and especially the math-oriented areas of computer science such as deep learning. So it is no accident that I selected *Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation* (Pragmatic Programmers) by Mark C. Chu-Carroll to come in at such a high spot on this list đŁ

Speaking of the relevance and pervasiveness of math throughout the fabric of science, I can’t help but draw parallels to, and resonate with, the marvelously erudite *and* approachable essay by the Hungarian-American theoretical physicist, engineer, and mathematician Eugene Wigner. It’s entitled *The Unreasonable Effectiveness of Mathematics in the Natural Sciences.*Â Don’t miss it đ

### No Royal Road to Deep Learning

Meanwhile, reminding ourselves of the adage that there is no royal road to geometryâor to *deep learning*, for that matterâa sustained study ofÂ *Good Math* will benefit you immensely. Here, then, are some reasons why I’m compelled to pass this judgment onÂ *Good Math*, starting with the fact that

*Good Math*Â is accessible to anyone with a basic high school background in math đ- To get the most out of it, you need nothing except curiosity đ
- It doesn’t have to be read cover-to-cover since each chapter is pretty much standalone đŻ
*Good Math*Â systematically builds your enthusiasm for math instead of boring you to tears đ- Trust me on the point above, especially, since you’ll need to ramp up your prowess in the area of handling mathematical machinery đ
- In turn, to the point above, in cascading fashion with the point that preceded
*it*, the barriers to entry are low when it comes to deep learning because you do notâI repeat, you do*not*âneed any esoteric math beyond good old linear algebra and optimization techniques to conquer deep learning đ *Good Math*Â conveys the joy of the mathematical landscape to the inquisitive mind đ

### A Bit on Coursera for Context

To put my thoughts above in context, let’s segue a bit. So I earned a certificate online last year from Courseraâspecifically a certificate for the *Machine Learning*Â (ML) course taught by Andrew Ng, then with Stanford Universityâwhich will make you do *deep* dives (pun was totally unintentional) through the ocean of linear algebra, so I have a *pretty* good idea of what I’m talking about in the bulleted list above đ

And while we’re talking about Coursera, I’ll mention in passing that they’ve got some of the most well-thought and well-designed courses available online; it was actually my passion for the Scala programming language which, by the way, brought me to Coursera in the first place. And that’s how I discovered their offering on ML and other cool stuff. Other courses I tookâand for which I also earned a certificate eachâinclude the following, which I can highly recommend, should anyone have an interest in this sort of thing:

*Functional Programming Principles in ScalaÂ**Functional Program Design in ScalaÂ**Parallel Programming*

### And Speaking of the Scala Programming Language

Scala, by the way, is incredibly well-suited for doing ML, but don’t get me even started there; suffice it to say that the definitive book on the confluence of Scala and ML is IMHO the one entitled *Scala for Machine Learning* (Packt Publishing) by Patrick R. Nicolas. I’ve made at least six passes through the heft of *Scala for Machine Learning*âall 400-plus pages in their glory with a highlighter in hand and a trusty Macbook at my sideâand *still*Â keep finding new nuggets each time I revisit its pages. Go figure đŁ

### Favorite Chapters

My favorite chapters inÂ *Good Math*Â happen to be consecutively located, and they are:

17. Axiomatic Set Theory: Keep the Good, Dump the Bad

17.1 The Axioms of ZFC Set Theory

17.2 The Insanity of Choice

17.3 Why?Â

18. Models: Using Sets as the LEGOs of the Math World

18.1 Building Natural Numbers

18.2 Models from Models: From Naturals to Integers and Beyond!Â

19. Transfinite Numbers: Counting and Ordering Infinite Sets

19.1 Introducing the Transfinite Cardinals

19.2 The Continuum Hypothesis

19.3 Where in Infinity?

### Honest and Intelligent Humor

Last, but not the least, the honest and intelligent humor that’s liberally sprinkled throughout *Good Math* is best revealed, I think, right at the outset of the book, in the *Dedication* section in fact, where we hear Chu-Carroll tell the reader that

This book is dedicated to the memory of my father, Irving Carroll (zt”l). He set me on the road to becoming a math geek, which is why this book exists. More importantly, he showed me, by example, how to be a mensch: by living honestly, with compassion, humor, integrity, and hard work.

### Judgment

Dig in, and you’ll see what I’m talking about.Â This is a book for someone who is looking for inspiration and gentle guidance on mastering mathematicsâand don’t we *all*? If that’s you, don’t missÂ *Good Math*Â đ

#### 3. *How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics* (Princeton University Press) by William Byers đ

### Let’s Start Conceptualizing

You won’t find a shred of linear algebra in the next book here, which is a superb one, by the way, and entitled *How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics* (Princeton University Press) by William Byers. You will, however, find *plenty* in its pages to get you thinking mathematically; that, I think you’ll agree, counts for a lot if you wish to equip yourself for tacklingâin terms of conceptualizingâthe mathematical machinery that underpins linear algebra. Again, absolutelyÂ *nothing* to dread: No rocket science going on here đ What you have here is simply stellar guidance on learning how to build beautiful precept upon precept upon precept, till you’ve built for yourself a grand conceptual edifice of magnificence and wonder âș đĄ đ Â đą đ°

To drive home the pointâthough at the risk of sounding like a broken recordâI should divulge that I’ve always been endearingly impressed by the onion metaphor of conceptualizing, the constant revisiting of basics for increasingly profounder conceptualizations. I trace this to my undergrad years when I spent countless hours poring over the mesmerizing pages of the classic MIT textbook entitled *Circuits, Signals, and Systems* by William M. Siebert đ

### Rave Review

So it’s no surprise that a distinguishedÂ mathematician and progressive thinkerâDavid Tall, coauthor of *Algebraic Number Theory* and *Fermat’s Last Theorem*âhad this to say aboutÂ *How Mathematicians Think*. In a rave review of the Byers book we’re looking at here, here was Tall’s call:

This is an important book, one that should cause an epoch-making change in the way we think about mathematics.

While mathematics is often presented as an immutable, absolute science in which theorems can be proved for all time in a platonic sense, here we see the creative, human aspect of mathematics and its paradoxes and conflicts. This has all the hallmarks of a must-read book (italics mine).

### Let’s Get Ourselves Acquainted

To help you get situated with the marvelously relevant scope of the contents of *How Mathematicians Think*Â for foundational mathematics, let’s take a peek at its table of contents:

INTRODUCTION

Turning on the Light

SECTION IâTHE LIGHT OF AMBIGUITY

CHAPTER 1:ÂAmbiguity in Mathematics

CHAPTER 2:ÂThe Contradictory in Mathematics

CHAPTER 3:ÂParadoxes and Mathematics: Infinity and the Real Numbers

CHAPTER 4:ÂMore Paradoxes of Infinity: Geometry, Cardinality, and Beyond

SECTION IIâTHE LIGHT AS IDEA

CHAPTER 5:ÂThe Idea as an Organizing Principle

CHAPTER 6:ÂIdeas, Logic, and Paradox

CHAPTER 7:ÂGreat Ideas

SECTION IIIâTHE LIGHT AND THE EYE OF THE BEHOLDER

CHAPTER 8:ÂThe Truth of Mathematics

CHAPTER 9:ÂConclusion: Is Mathematics Algorithmic or Creative?Notes

Bibliography

Index

I’ve readÂ *How Mathematicians Think*Â at least four times since acquiring it a few years ago; it’s given me the joy of tackling the heart of conceptual mathematical machinery in a way *no* other book has, imbuing me with renewed vigor to fearlessly pursue the guts of linear algebraâI thank Byers for giving us this gem đ

### Favorite Chapters

While the whole book is marvelous, I found myself especially reveling in the awesomeness of two standout chapters:

- CHAPTER 5:Â
*The Idea as an Organizing Principle*) đ„ - CHAPTER 9 (
*Conclusion: Is Mathematics Algorithmic or Creative?*) đŠ

### Two Other Books by the Author

So I’ve also got two other books by Byers as well, and one of themâ*Deep Thinking: What Mathematics Can Teach Us About the Mind*âis also just plain awesome. I *nearly* included it in this essay as well. Alas, I simply didn’t have enough time to do justice to its content, and had to reluctantly jettison it off for now; perhaps it’ll have to wait for a future installment in this series of essays on deep learning đł

### Is Math Algorithmic or Creative?

There are so many lovely passages I wished to share from the pages ofÂ *How Mathematicians Think*Â that I had a hard time selecting one. Plus we would’ve digressed; then again, I remind myself of the name of this blog, *Programming Digressions* đ So let’s look at just this one brief passage in which Byers is proposing to the reader an intriguing questionâat the outset of CHAPTERÂ 9 (*Conclusion: Is Mathematics Algorithmic or Creative?*)*Â *to be preciseâin a thoughtful style that’s the hallmark of this book:

I am asking whether it is conceivable that at some time in the future computers could completely take over the show, whether a machine could be programmed to “do” mathematics from start to finish. This would involve (among other activities) examining mathematical situations or situations that potentially could be mathematized, producing data about these situations, generating conjectures, and demonstrating the validity or invalidity of these conjectures.Put in this way, the answer to the question of whether a computer could ever do mathematics is clearly “No!”(The discussion about whether a computer can do mathematics is usually restricted to the last of these activities, namely, demonstrating the validity of certain mathematical statements.) (italics mine)

### Judgment

And speaking of the sentence above in italics, that’s where youâyes, you, future deep learning expertâcome in. If I were you, I wouldn’t missÂ *How Mathematicians Think*Â for the life of me. Enough said đŹ

#### 4. *Matrix Analysis and Applied Linear Algebra* (SIAM: Society for Industrial and Applied Mathematics) by Carl D. Meyer đŹ

### This Books is a Thorough Workhorse

You have a true workhorse of a book in this one which is entitledÂ *Matrix Analysis and Applied Linear Algebra* (SIAM: Society for Industrial and Applied Mathematics) by Carl D. Meyer. The guts of linear algebra are laid bare in its pages unlike in any other book on the subject. Yes, I still stand by my earlier assessment that the book by Poole (*Linear Algebra: A Modern Introduction*) is matchless when it comes to helping you build, strengthen, *and* elevate yourÂ foundationalÂ skills to pursue deep learning as far as you wish toâwhich is why it came in at the top spot in our listâthe undeniable fact is that you’ll also benefit from being exposed to yet another perspective, or two đż

### Tons of High Quality Worked-out Examples

AndÂ *Matrix Analysis and Applied Linear Algebra*Â is simply stellar in its clarity, rigor, *and* friendly style of explaining everything. It’s got tons and tons of nicely worked-out examples of linear algebra problems. Okay, rest assured, this book is *not* written in the drive-by style of something like a book in theÂ Schaum’s outline series, decent though the books in that series are, filling a useful gap in disseminating knowledge at a mechanical level.

So if your learning style leads you in the direction of seeking an examples-driven approach, you’re going to loveÂ *Matrix Analysis and Applied Linear Algebra*Â đ

### A Bit on MIT Technology Review for Context

As a long-time subscriber to the inimitably brilliant magazine *MIT Technology Review*Â (affectionately know in tech circles simply as *MIT TR*), I had cited a brief excerpt from one of its issues elsewhere, noting how, when you program in Clojureâas you would with any Lisp dialectâyou get to define your own domain-specific languages (DSLs) to suit your needs. I had also talked about how there’s no conceptual burden whatsoever in that context, because you become a language *designer*, making your own DSLs through which you encode exactly how your specific business use cases work. But we digress đ»

*did*wish to cite fromÂ

*MIT TR*Â now, though, is an incisive take by Nanette Byrnes on the pervasive applications of deep learning to seemingly limitless areas of business. Byrnes, who is

*MIT TR*‘s senior editor for business, reflects on the state of the union in the AI universe in general, and the deep learning world in particularâin her editorial comments in a fairly recent issue of

*MIT TR*(July/August 2017 Vol. 120, No. 4)ânoting cogently how đ

Artificial intelligence, a technology that we believe will drive much of the economic growth over the next few years, is sprinkled throughout the list. The company at the top, Nvidia, has gained expertise in AI and used it to transform itself. Once known as a maker of chips for gaming, it is now a leading player in deep learning and autonomous vehicles. Amazon, No. 3, is on the list again for its ambitions to build an AI-powered store and place the technology at the heart of the home of the future.

### How AI and Deep Learning are Shaping the World

And while we’re still on the subject of the undeniably brilliant coverage by *MIT TR* of stuff related to the future of technology, basically where technology is taking us nextâtrust me, they are definitely *not* making idle boasts when they claim that the mission of *MIT TR* is to equip its audiences with the intelligence to understand a world shaped by technologyâlet’s listen to an even *more* intriguing pointer on a topic that’s decidedly germane to our deep dive into, um, the ocean of deep learning đł

*already*! đ± đ

### Conceptualizations for Evolving Views of Business

And godspeed, invoking a phrase (i.e. “make it snappy”)Â that I recently used elsewhere, I offer to you these prescient words of Tom Simonite as he shares an intriguing pointer to unraveling the significance of deep learning to transforming our evolving worldview of business. So let’s now hear Simonite as he opines on how Google thinks it can wrest the cloud computing market away from Amazon by helping companies make use of machine learning. Thus, in a nice wrap-up of how AI and deep learning are propelling Google into yet higher stratospheres of technological prowess, Simonite reminds us in a fairly recent issue of *MIT TR* (July/August 2017 Vol. 120, No. 4) how đ

Early in 2015, artificial-intelligence researchers at Google created an obscure piece of software calledTensorFlow. Two years later the tool, which is used in building machine-learning software, underpins many future ambitions of Google and its parent company, Alphabet.TensorFlowmakes it much easier for the companyâs engineers to translate new approaches to artificial intelligence into practical code, improving services such as search and the accuracy of speech recognition. But just months afterTensorFlowwas released to Googleâs army of coders, the company also began offering it to the world for free.

### Advice

Is that intriguing or what? If you want to have a piece of the pie, the advice I have for you is to get started with busting your linear algebra chops, which, in turn, will take you as far as you wish to go in the *land*âwait, I may have got my metaphors messed up a bit hereâthe *ocean* of deep learning đ

### Uncompromising Commitment

Again, the uncompromising commitment to help the reader conceptualize what exactly makes linear algebra tick is evident and writ large throughout the pages ofÂ *Linear Algebra: A Modern Introduction.* Meyer nicely sums up the essence of his fine bookÂ in an introductory section entitled *Purpose, Gap, and Challenge* by reminding us that

The purpose of this text is to present the contemporary theory and applications of linear algebra to university students studying mathematics, engineering, or applied science at the post-calculus level. Because linear algebra is usually encountered between basic problem solving courses such as calculus or differential equations and more advanced courses that require students to cope with mathematical rigors,

the challenge in teaching applied linear algebra is to expose some of the scaffolding while conditioning students to appreciate the utility and beauty of the subject. Effectively meeting this challenge and bridging the inherent gaps between basic and more advanced mathematics are primary goals of this book (italics mine).

### Let’s Get Ourselves Acquainted

To help you get situated with the scope of the contents ofÂ *Matrix Analysis and Applied Linear Algebra*Â and their relevance to foundational mathematics, let’s take a peek at its table of contents đŻ

Chapter 1:

Linear Equations

Chapter 2:Matrix Algebra

Chapter 3:Vector Spaces

Chapter 4:Norms, Inner Products, and Orthogonality

Chapter 5:Determinants

Chapter 6:Eigenvalues and Eigenvectors

Chapter 7:PerronâFrobenius Theory

### A Truly Standout Chapter

While the whole book is rock solid, I found myself especially resonating with the content of the following standout chapter:

5.

Norms, Inner Products, and Orthogonality

5.1 Vector Norms

5.2 Matrix Norms

5.3 Inner-Product Spaces

5.4 Orthogonal Vectors

5.5 GramâSchmidt Procedure

5.6 Unitaryand Orthogonal Matrices

5.7 Orthogonal Reduction

5.8 Discrete Fourier Transform

5.9 Complementary Subspaces

5.10 Range-Nullspace Decomposition

5.11 Orthogonal Decomposition

5.12 Singular Value Decomposition

5.13 Orthogonal Projection

5.14 WhyLeast Squares?

5.15 Angles between Subspaces

*all*experts were beginners once đ¶ đŠ đš đŽ đ

### Wrap Up

*dive*into this rock solid bookâwait, wait, I think I got my metaphors messed up big time hereâbecause a dive into a book that’sÂ

*rock*solid might, um, be rather painful đ So let’s rewind a bit and rephrase all that and instead say something like: Let’s wrap up our

*drilling*into this rock solid bookânow

*that*sounds better, doesn’t it?âby checking out what Meyer has to say in another introductory section entitledÂ

*Comprehensiveness and Flexibility*. There, the author ofÂ

*Matrix Analysis and Applied Linear Algebra*Â reminds us that

A rather comprehensive treatment of linear algebra and its applications is presented and, consequently, the book is not meant to be devoured cover-to-cover in a typical one-semester course. However, the presentation is structured to provide flexibility in topic selection so that the text can be easily adapted to meet the demands of different course outlines without suffering breaks in continuity.

Each section contains basic material paired with straightforward explanations, examples, and exercises. But every section also contains a degree of depth coupled with thought-provoking examples and exercises that can take interested students to a higher level.The exercises are formulated not only to make a student think about material from a current section, but they are designed also to pave the way for ideas in future sections in a smooth and often transparent mannerÂ(italics mine).

### Judgment

And there you have itâmuch as I said at the outset, you have in *Matrix Analysis and Applied Linear Algebra* a true workhorse of a book đ

#### 5. *Thinking Mathematically* 2nd Edition (Pearson) by J. Mason, L. Burton, K. Stacey đĄ

### Boost Your Mastery of Mathematical Thinking

Last, but *certainly* not the least, is another remarkable book that will boost your mastery of mathematical thinking in an effortless and delightful way: *Thinking Mathematically *(*2nd Edition*) (Pearson) by J. Mason, L. Burton, and K. Stacey. I’m not aware of a kinder, gentler approach to introducing yourself to a study of theÂ foundational aspects of mathematics that will, in turn, prepare you to take on linear algebra in earnestâlet’s also remind ourselves that linear algebra, *in turn*, is your ticket to the exciting world of deep learning. And I’m doing my bit here to lower the barriers to entry đ§ đ„ đŠ

### The Idea of Learning by Wholes

Let’s put it this way:Â *Thinking MathematicallyÂ *is probably what David Perkins has in mind whenâin his indispensable and highly readable book entitledÂ *Making Education Whole*Â (Jossey-Bass)âhe talks about the concept of “learning by wholes”. That’s the central theme of *that* book where Perkins tells us in his engaging style that

…this is what learning by wholes is all about. Learning by wholes aims squarely at learning from the lively now.Â

Its goal is to build learning out of endeavors experienced as immediately meaningful and worthwhileâjunior versions of the whole game that build toward more sophisticated versions.Â Its commitment is to leverage features of good naturalistic learning, whether we are talking about Bierstadt, baseball, or Barcelona. Its method is to systematize important features of such learning through the seven principles. Its credo says that good learning is learning from a richly experienced today with tomorrow in view (italics mine).

### AI and Learning by Wholes

If you buy that philosophyâand frankly, it’s an eminently sensible oneânotwithstanding that Perkins, the author of the aforesaid philosophy, has impeccable credentials. In fact, in his own words, Perkins shares this intriguing insight by way of background, telling the reader how

My academic degrees are from the Massachusetts Institute of Technology (MIT). I was a mathematics major. After I finished the undergraduate work, I continued into a doctoral program, developing an interest in mathematical approaches to artificial intelligence. Artificial intelligence is the study of how to get computers to undertake intelligent activities, such as playing chess or proving mathematical theorems or controlling a robot to do interesting and challenging things.

My work on artificial intelligence stimulated my interest in thinking and learning in human beings.After finishing my degree, I slid over into the world of cognitive psychology and education, but the why of that is another story. Right now, you can picture me in the foothills of the dissertation range, thinking about what kind of research on artificial intelligence to attempt (italics mine).

### A Brilliant and Friendly Guide to the Mathematical Process

Having made the case, I suppose, for the rightful place of a holistic approachâone marked by profound respect for beauty in all its shapes and formsâI’ll add that there is perhaps no better way to introducing the essence of *Thinking Mathematically* than by way of the following brief section from the book’s *Introduction* where the authors tells us that

Thinking Mathematicallyis about mathematical processes, and not about any particular branch of mathematics. Our aim is to show how to make a start on any question, how to attack it effectively and how to learn from the experience. Time and effort spent studying these processes of enquiry are wisely invested because doing so can bring you closer to realizing your full potential for mathematical thinking. Â

Experience in working with students of all ages has convinced us that mathematical thinking can be improved by

- tackling questions conscientiously;
- reflecting on this experience;
- linking feelings with action;
- studying the process of resolving problems; and
- noticing how what you learn fits in with your own experience.

### How to Use This Book Effectively

Then, farther along in the book, in a subsection entitled *How to use this book effectively!*Â to be precise, the authors pointedly tell the reader that

Thinking Mathematicallyis a book to beusedrather than read, so its value depends on how energetically the reader works through the questions posed throughout the text…Probably the single most important lesson to be learned is that being stuck is an honourable state and an essential part of improving thinking.However, to get the most out of being stuck, it is not enough to think for a few minutes and then read on (italics mine, and BTW, “honourable” isnotÂ misspelled; all I can say as an American đ Â is that it’s just our Aussie friends đ getting carried away đ with spelling things in their own wayward way đ).

Dare I add that, after reading the brief excerpt above, we were all vigorously reassuring ourselves, saying things like the following to ourselves, “Sure, *other* people may get stuck while learning. But *us* technologistsâ*we* uber hacker types who chew and spit functors, monads, CQRS and stuff as if they were watermelon seedsâ*we* never get stuck, now do we ever? YMMV đ

### Judgment

Don’t miss this book! I’ll add only this much that if you read *Thinking Mathematically*Â carefully, you can’t help but see the intellectual imprint of the legendary George PĂłlyaâthe Stanford University mathematician who enlightened the world with the then-groundbreaking concept of *heuristics*âwrit large in its pages. In fact, the authors of this book graciously acknowledge as much, noting in an introductory section entitled *The Power of an Experiential Approach* how

The original book was conceived as an exposition of our own experience as mathematical thinkers, profoundly influenced by the work of George PĂłlya.

#### Conclusion âș

Until we meet next time, in *our* next essayâhey, I hasten to add that I’m positively not using the royal “our” there and instead simply indicating that this is *our* blog, and not mine aloneâI’ll slip in this meme edgewise: About the pic above, the one that adorns this book’s review, my copy ofÂ *Thinking Mathematically*, sleek black color and all, finds itself valiantly propped up against the computer monitor standing across from my Mac keyboard đ

*Thinking Mathematically*Â literally fell apart at the seams, shall we say, from extended use; the careful reader will note in the pic above how my copy of the book spiral-bound. As they say, some books fall into

*disuse*; others into

*use*; and yet others into

*overuse*đ»Â

#### An Invitation đŁ

In the end, and much as I hinted at, at the very outset, I invite your commentsâHaving now read the brief take each on the books above đ€

- Do you find that your experience of reading any of these books was different? đą
- Did I perhaps not cover some qualities, and which are the ones that you actually found the most helpful as you learned, um, deep learning and its ecosystem? đ
- Did I leave out any of your favorite deep learning foundational books? đ

My hope is that these brief vignettes will help you in your journey to grokking deep learningâI leave you with a fleetingly brief pic collage below this time; a more elaborate pic collage awaits you in a future essay just around the corner đ

#### Dedication đ

This essay somehow got me resonating with two songsâboth by Bryan Adamsâso I’ve got lyrics from those songs as well to accompany the pic collage below, *each* of the two dedicated to my better half, with love đÂ

*much*Â deeper things. Among other things, she has leavened my pragmatism by imbuing me with the primacy of humbleness, and helping everyone around us grow and blossom. Darling, this essay is for you đż

#### Bon Voyage đąÂ

And to you, as you embark on the journey of a lifetime, setting sail on the sea of deep learning, I say with much warmth and friendship, *Bon voyage* đŹ

#### Collage of Pics and Lyrics đž

I’d like to see you, thought I’d let you knowI wanna be with you everydayCause I’ve got a feeling that’s beginning to growAnd there’s only one thing I can sayÂ

I’m ready – to love youI’m ready – to hold youI’m ready – I’m readyReady as I’m gonna beÂ

She left me a long note when she left me hereTold me that love was hard to findBut baby it’s easy and I’ll make it clearThat there’s only one thing on my mindÂ

I’m ready – to love youI’m ready – to hold youI’m ready – I’m readyReady as I’m gonna be

~ Bryan Adams (Lyrics fromÂI’m Ready) đ

The biggest lie you ever told – your deepest fear ’bout growin’ oldThe longest night you ever spent – the angriest letter you never sentThe boy you swore you’d never leave – the one you kissed on new years’s eveThe sweetest dream you had last night — your darkest hour, your hardest fightÂ

I wanna know you – like I know myselfI’m waitin’ for you – there ain’t no one elseTalk to me baby – scream and shoutI want to know you – inside outI wanna dig down deep – I wanna lose some sleepI wanna scream and shout – I wanna know you inside outI wanna take my time – I wanna know your mindYa know there ain’t no doubt – I wanna know you inside outÂ

The saddest song you ever heard – the most you said with just one wordThe loneliest prayer you ever prayed – the truest vow you ever madeWhat makes you laugh, what makes you cry what makes you mad, what gets you byYou highest hight, your lowest low – these things I want to knowÂ

I wanna know you – like I know myselfI’m waitin’ for you – there ain’t no one elseTalk to me baby – scream and shoutI want to know you – inside outI wanna dig down deep – I wanna lose some sleepI wanna scream and shout – I wanna know you inside outI wanna take my time – I wanna know your mindYa know there ain’t no doubt – I wanna know you inside outÂ

I wanna know your soul – I wanna lose controlC’mon n’ let it out – I wanna know you inside outYa gotta dig down deep – I wanna lose some sleepI wanna scream and shout – I wanna know you inside outTell me everything..

~ Bryan Adams (Lyrics fromInside Out) đ§

“The challenge in teaching applied linear algebra is to expose some of the scaffolding while conditioning students to appreciate the utility and beauty of the subject.”We don't normally don't include the words 'art' or 'beauty' within STEM subjects and it is refreshing when I read about these disciplines in that light. The mystery, the wonder, the beauty, the awe-inspiring realizations, the creative process, the faith, the unquantifiable experiences are inextricably woven into the fabric of scientific and mathematical scholarship. How can we deny this?I'm reminded by Professor Abdus Salam's acceptance speech as he received his Nobel Price:”Thou seest not, in the creation of the All-merciful any imperfection, Return thy gaze, seest thou any fissure. Then Return thy gaze, again and again. Thy gaze, Comes back to thee dazzled, aweary.” – the Holy Quran.I'm reminded also of music. The study of music and music theory is considered a fine art but is also very mathematical. The traditional notion that art and creativity vs science and rational thinking exist in two separate hemispheres is far from reality. I'm often asked how I can be a painter and a physician simultaneously as if the two are a peculiar combination. In my mind, those two overlap beautifully.Fascinating essay, Akram Mamu. I admit (as someone who has never studied engineering) some of the language is foreign to me but I quite enjoyed the meat of it (as well as your fun digressions and quotes of lyrics đ.)

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– Thank you, Saira beti, for sharing

your profound impressions on reading this essay. I'm delighted that you found it enjoyable. In your own gracious words: “Fascinating essay”!- While I remain, as ever, a hard-core technologist who has been and remains in the trenches of designing, architecting, and indeed crafting software for a bit over two decades,my work has at the same time been informed and deeply influencedby theseeminglyethereal qualities of taste and beauty. In these particular aspectsâthe pursuit of taste and beauty in matters of designing software and crafting proseâprobably the biggest influence on my work has been that of Paul Graham –Many more details than I could fit herein my response can be found on exactly that topic in an essay elsewhere.- Here, I present ony abriefexcerpt that I lifted from that essay: “So I got to thinking about something that the noted Lisp hackerâand founder of Y Combinator, now the preeminent startup-company acceleratorâPaul Graham had to say in connection with the pursuit of beauty in design. So I dug up my well-worn copy of Hackers & Painters (O'Reilly Media), harking back to my days in wintry Minnesota â and boy, was I surprised, pleasantly so, I hasten to add, to rediscover, in revisiting its pages, just how profound and lasting an impact the pursuit of beauty has in creating and evolving designs of the highest order. To give you a better flavor of Graham's take on this subtle matter, I really can't do much better than share some excerpts as I flip through the well-worn (wizened? although using this latter word would be a tad too anthropomorphic) pages of Hackers & Painters…”- As someone who is fortunate enough to know aboutthe genius in the art that you create, I think that the world at large should know about your amazing work. I believe thatyour voice speaks for a new generationwhich looks for inspiration in the timeless qualities of honesty, truth, and integrity… So it is with great pride that I share here a link to give other readers a sense for some of the amazing work you've done andcontinueto do!- I've actually got a bunch of ideas in connection with your profound comment above,and will return and add to this response unhurriedly;it does not serve anyoneâleastof all you and my other readersâto get a drive-by, hurriedly written response from me. So I ask you all to please stay tuned – In the immortal catchphrase associated with Arnold Schwarzenegger,“I'll be back” đLikeLike

– I now fulfill the promise that I had made in my previous comment, saying

that “I'll be back” đ– Here, then, are the ideas that were percolating in my mind after reading your profound comment. In no particular orderâother than the order in which they appeared in your commentâI've got here a (relatively) unhurried response…[ONE]To your first point (“We don't normally don't include the words 'art' or 'beauty' within STEM subjects and it is refreshing when I read about these disciplines in that light“), I trust that my previous responseâwhere I sought to articulate how my work has always been deeply influenced by the apparently ethereal and admittedly intangible qualities of taste and beautyâhelped to reaffirm similar thinking on my endregarding why it's vital to include memes like 'art' and 'beauty' within STEM subjects…[TWO]To your second point (“The mystery, the wonder, the beauty, the awe-inspiring realizations, the creative process, the faith, the unquantifiable experiences are inextricably woven into the fabric of scientific and mathematical scholarship. How can we deny this?“), I couldn't agree more vigorously![THREE]To your third point (“I'm reminded of Professor Abdus Salam's acceptance speech as he received his Nobel Prize…“), the verse you quote is sublimely expressive, andits message of splendor rings truein unison with what little I know about our elegant universe. I can't help but thinking to the inspiring words of Sir Isaac Newton, the English mathematician and physicist (1642 – 1727) when he noted how:”I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.“Speaking of genius,I'll add a personal anecdoteregarding Professor Abdus Salam, a towering figure in 20th century theoretical physics who shared the 1979 Nobel Prize in Physics with Sheldon Glashow and Steven Weinberg for his contribution to the electroweak unification theory, and whom you mentioned in your comment. So my late grand uncle Naseer Ahmad Faruquiâa distinguished scholar who earned his education at Cambridge University in Englandâonce had the honor and privilege of hosting Professor Salam at his home, many years ago. He told me vivid recollections of that meeting, and how deeply he was impressed by Professor Salam's profound insights into the intellectual fabric of our universe.LikeLike

[FOUR]To your fourth point (“I'm reminded also of music. The study of music and music theory is considered a fine art but is also very mathematical. The traditional notion that art and creativity vs science and rational thinking exist in two separate hemispheres is far from reality.“), I find myself agreeing with your assessment. Yep,the dichotomy (between art and science) that has foisted upon the gullible publicâand I'm not by any means attributing any malicious intent to anyoneâis a patently false one. And since you thoughtfully mentioned about the “two separate hemispheres”, I'll mention in passing the name of an interesting bookâThe Master and His Emissary: The Divided Brain and the Making of the Western(Yale University Press)âshould anyone be interested in pursuing this theme further. – While fascinating, that book is a bit of a boat anchor, and I've made only a small dent into its heft of 544 pages. At any rate, the author ofThe Master and His Emissary, Iain McGilchrist, talks in great details aboutyet another confounding dichotomy, that one between our two brain-hemispheres. In the words of Mary Midgley withThe Guardian, McGilchrist “looks at the relation between our two brain-hemispheres in a new light, not just as an interesting neurological problem but as a crucial shaping factor in our culture…splendidly thought-provoking…” FWIW, I learned about this book by chance through the pages of an equally remarkable book entitledThe Shadow of the Tsunami: and the Growth of the Relational Mind(Routledge) by Philip M. Bromberg, Ph.D. More details on that in an essay elsewhere…[FIVE]To your fifth point (“I'm often asked how I can be a painter and a physician simultaneously as if the two are a peculiar combination. In my mind, those two overlap beautifully.“), all I can say is that I have a soft spot in my heart for precisely this kind of “blending”.I refer you and interested readers to two additional resources:(1) An essay entitled “When Object Orientation Met Functional Programming” where you can read up on how two seemingly disparate programming philosophies can overlap beautifully and “blend” in fact , and (2) Some resources I've got with further links for exploration.- Suffice it to say that I went one step further with “blending” ; although presented in an admittedly technical context, I think that the discussion in which it is embedded, not to mention its scope, is broader, transcending the false boundaries that were erected by someone at some time and they continue to persist as happenstance. That essay can easily be viewed in a larger context.- I think it's high time to break down such false boundaries that wall some in and wall some out. Remember the line “Tear down this wall!” from a speech made by US President Ronald Reagan in West Berlin, calling for the leader of the Soviet Union, Mikhail Gorbachev, to open up the barrier which had divided West and East Berlin since 1961?Perhaps we need a similar rallying cryin the dimension of the arts and sciences… I invite your thoughts as well as, of course, the thoughts of other readers.LikeLike

[SIX]To your sixth point (“…I admit (as someone who has never studied engineering) some of the language is foreign to me but I quite enjoyed the meat of it (as well as your fun digressions and quotes of lyrics.“), let me assure you thatthe foreignness which you discernedin (at least some of) the language I used in this essayâthe gradual preamble at the outset clearly led us into some decidedly technical materialâis a perfectly natural reaction. I always try to leaven technical material with some fun along the way; as humans, we can condition ourselves to digesting onlysomuch hard-core technical material before our human nature begins to balk and start sounding notes of vociferous protestations :)-I should hasten to point out an important distinction:Let's not confuse “accidental complexity” with “inherent complexity”.Conflating these two kinds of complexitieswill inevitably lead us to troubling ends. Here I almost hear the admonition of my English teacher who was wont to tell usâhis captive audience of high schoolersâthat a word such as “complexity”, being an uncountable noun, should never be used beyond its singular sense under any circumstances. That's right, notwithstanding extenuating circumstances. But then again, my esteemed English teacher is not around to admonish us,so we'll blithely use uncountable nouns in all their glorious plurality đ– Reminds me of how Winston Churchillâwhen supposedly an editor had clumsily rearranged one of Churchill's sentences to avoid ending it in a prepositionâwho was very proud of his style, scribbled this note in reply: “This is the sort of English up with which I will not put.” ;)- Glad you enjoyed “the fun digressions and quotes of lyrics”.Readers have grown accustomed to my proclivity for digressing. Wherefrom the penchant came and wherefore I submit to the propensity, I really can't say because, frankly,I myselfdon't know the answer either! What Icansay with some confidence is that readers apparently enjoy these excursions and explorations. More fundamentally,that just is my style of writing, which is not changing anytime soon ;)- Oh, and I digress in my essays on a routine basis not because I'm feeling feisty all the time or, heaven forbid, that I perversely want to disorient my readersâinstead, I assume that the reader has a million other more interesting things to do with their time and is just waiting for an excuse to tune out. Yep, if anything,I go one step further in thatI've aligned myself with the reality on the ground that's far more along the lines of what the American writer Kurt Vonnegut probably had in mind when he used undeniably visceral language in telling his readers, in the first person of course, how:”When I write, I feel like an armless, legless man with a crayon in his mouth.“LikeLike

–

Incidentally, the Vonnegut quoteabove happens to be one that was trying to make its way into thefirstdraft of the essay, but it kept getting pushed farther down in the draftâin the style of push-down automata or simply our trusty stack data structureâuntil I couldn't make it fit anywhwere, and then it completely fell off of the essay :(- Finally, I offer to youâprovided, of course, that anyone isa concluding thought, which is that when I began writing essays for this blog, I had an audience of exactly one: me. I recall, too, a sentiment that I had echoed a long time agostillawake lolââit's a sentiment that still rings true with meâthat I would continue to write even if I had an audience of only one.- So “I write (prose) to better understand what I think“, which is probably a variant of the adage I read somewhere a long time ago that “I write code to better understand what I design“… Whathaschanged during the intervening years is thatthousands of readers now come to this blog every monthto read essays such as this one, typically on a cadence of an essay every weekend. – Slowly but surely, the responsibility of honoring the time of so many readers has irrevocably altered my view of how an essay ought to be written. It has not, and nothing ever will, alter my writing style!Thomas Jefferson once memorably said, “In matters of style, swim with the current; in matters of principle, stand like a rock.” Yep, Jefferson definitely struck a chord with me there… Today, I care atleastas much foryourreading experience as I did for my own when I was the solitary reader of the blog. Whether I succeedâor failâyou all have been officially designated as the judges,so please pronounce away your judgements by way of comments đLikeLike

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Great essay Akram, the new formatting is much better now ;)Is Linear Algebra a important pre-requisite for standard Machine Learning ?Do you think it's helpful in having a foundation in standard ML before attacking DL ?If you had to choose one book from Good Math or the Thinking Mathematically books what would you decide on ?

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– Thank you, Revanth, for making the time to read this essay, plus sharing (your training offerings) via the comment…- While I don't mind at all when readers share links (to their training offerings, etc.), I would ask that you also please contribute to the discussion by sharing your thoughts, observations, and how we can make this a better blog for all readers, thanks đ

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– Thank you, Shahbaaz, for making the time to post your comment, sharing those warm words of encouragement,

along with asking some really great questions!– To your comment (“Great essay Akram, the new formatting is much better now đ“), I say, thank you! Glad you enjoyed the content of the essay; regarding the presentationâas I've come to realize over the yearsâit does count for quite a bit, too. After all, a nice presentation (breaking the “wall of text” into easily digestible chunks, etc.) makes your reading experience that much more pleasant. I'm always look for ways to improve the blog, so please keep those suggestions coming :)- As to your first question (“Is Linear Algebra an important pre-requisite for standard Machine Learning?“)âthat's an excellent question because it naturally arises in the mind of most beginnersâI would say that, strictly speaking, you don't need to learn linear algebra before you get started in machine learning. But to dive deeper, you can't get around much without a solid grasp of linear algebra concepts. That ties in, in fact, rather nicely with your next question,so I'll answer part of it here:– So while you don't need to learn linear algebrabeforeyou get started in machine learning, strictly speaking, you can't get anywhere in “Deep Learning” without a solid mastery of linear algebra. That's precisely why two of the five stellar books which made the list in this essay are wholly devoted to linear algebra; I was going to add a couple moreâyep, you got it, on good old linear algebraâbut decided against it, lest it scare away too many readers ;)- And that's why I selectedsome other, non linear algebra (not non-linear lol), gentler math booksâimportant though they definitely areâinstead so as to leaven your experience and round it out. Rounding out your experience is a great idea in general, any way :)- To your second question (“Do you think it's helpful in having a foundation in standard ML before attacking DL?“)âwhich is another great questionâmy opinion is that it reallydoeshelp to see the larger context in which Deep Learning operates. But you don't have to tackle the two insequence; instead, feel free to take both on at the same time. If you do have the luxury to tackle the two sequentially, then by all means take on “Standard ML” first, and then move on to DL!- Finally, to your third question (“If you had to choose one book from“), my answer might surprise you and other readers: I would say, definitely go for the latter (Good Mathor theThinking Mathematicallybooks what would you decide on?Thinking Mathematically). I say this not becauseGood Mathisn't an excellent book. And since my answer will likely surprise youâas well as other readersâI owe you a longer explanation, especially given the ranking each I gave toGood Math(#2) and toThinking Mathematically(#5). I stand by everything I said aboutGood Math(in a nutshell,“This is a book for someone who is looking for inspiration and gentle guidance on mastering mathematicsâand don't we all?”) and toThinking Mathematically(in a nutshell, it's a “…remarkable book that will boost your mastery of mathematical thinking in an effortless and delightful way”). Allow me to add that neither of these would qualify as my desert-island-book, but that's a much larger dialog and digression đ –I love how readers are engaged with the blog, thanks!!!Do keep those questions coming!!!LikeLike

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– Thanks, Revanth, for sharing your insights into how Splunk makes machine data reachable, utilizable and helpful to everyone!- My experience is not with Splunk, but rather with the ELK stack of software tools (Elasticsearch, Logstash & Kibana), which are quite awesome… I'll take a look at SplunkâI've been hearing good things about Splunk over the years…- As you read more aboutÂ the top five

deep learningfoundational books, I hope it will inspire you and others to find more about how tools like Splunk,Â Elasticsearch, Logstash, andÂ Kibanacan help with Deep Learning endeavors:The only limit is our imagination đLikeLike

Another excellent post Akram. I very much enjoyed how you laid out the mathematical foundations of deep learning. I was encouraged to take another look at linear algebra (I dropped this class in college, along with numerical analysis, much to my disappointment later in life when I begin seeing it's application). I easily found the Deep Learning book by Goodfellow, Bengio, and Courville online for free and skimmed a few chapters. I think the books you covered in this post would be excellent primers to work thru before tackling the deep learning book in earnest.Keep up the good work my friend!

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Interesting to see what options are out there. I've had a passing interest in AI for a long time, but I only recently started actively writing DL-related code. There is a ton of information out there but most of it is either borderline superstitious (I read an article that claimed “nobody really knows why NN training works”) or so academic as to be impenetrable for beginners like myself.I'll definitely look into “Good Math” as linear algebra has never been one of my strong suits — I'm very much a “hands-on” learner, and having a simple guide to reference as I'm actually -DOING- would help my retention enormously.On another tangent, your opening remarks re: language purists put a funny thought in my head (especially WRT the way programmers tend to treat the “new kid on the block”): isn't it funny that even computer scientists can be virtual Luddites?

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– I enjoyed reading your thoughtful and detailed comment, Brendan, thanks!- To your first point

(“There is a ton of information out there but most of it is either borderline superstitious”)I couldn't help but smile and nod in agreement :-)- Yes, the subject of Deep Learning (DL) is, well,deeplymathematical at its foundations; over the years,a diverse bunch of people have made attempts, some were enlightened and rightly guided while a whole lot more were merely groping in the dark, so I'mnotsurprised that (during your commendable researches) you ran into “a ton of information out there but most of it is either borderline superstitious most of it is either borderline superstitious“!!- Don't despair though: Between my three write-ups on DL (right here on this blog that is “Programming Digressions”), this being the most mathematical of the three, I think I've got you covered.- So as to sidestep the unsettling aspects of a lot of theotherinformation out there which, as you very rightly noted, isâacademic as to be impenetrable for beginners like myself”, I highly recommend that you start with the 50,000 foot view that you'll find in the third of my three essays (writeups) on DL. It is super user-friendly, and I'm willing to wager that you'll find nary a chance of getting bogged down in technicalities:Best Deep Learning Books (Popular)– (Maybe I should have made that essay be the veryfirstone in the three-essays series on DL)- Cool, I think you're going to like the fine book calledGood Math, which is written by a fellow geek; it'sbya fellow computer scientist,forfellow computer scientists!- I'm with you: I, too, am very much a “hands-on” learner. My suggestion in this area would be to make the following book your best friend:Deep Learning: A Practitioner's Approach(O'Reilly Media) — The natural response to the preceding statement could well be,How would I do something like that?– Well, so I've covered exactly that (in depth) inthe second of the three essays right here, so check it out! The other book, which I also delve into deeply in that essay, is admittedly a bit academic, though awesome: it's bythe “three amigos” of the DL world, so you can put that one down as a (2nd) follow-up book to read up. Cool?- The concluding thought in your marvelous government especially caught my attention: “On another tangent, your opening remarks re: language purists put a funny thought in my head (especially WRT the way programmers tend to treat the “new kid on the block”):“- Excellent observation!!- BTW, you write with such grace that I feel compelled to invite you to pleaseisn't it funny that even computer scientists can be virtual Luddites?write a guest essayfor this blog sometime The most recent guest essay, of course, and which was on anentirelydifferent subject, was the one namedPop's War: My Father, the CIA, and the Green Death.LikeLike

Akram,Thanks for the compliment! As it comes from someone who somehow manages to make his writing both prolific and cerebral, I'll take it as high praise :)Perhaps I'll dig up one of my short stories some day and share it — or if I ever manage to get anywhere with this particular experiment (chasing NLP in the hopes of creating a passable chat bot) then I would be pleased to have the opportunity to share my thoughts on it. I'm a veteran programmer (13 years give or take) but as a total newbie in the field of deep learning I may have some observations/tribulations that veterans would find amusing.I'll definitely be ordering a few of the books you've cited here, too. I've yet to start on the other two essays you mentioned, but I'm sure they'll be just as informative. Programming is a humbling profession, and especially in the age of information I repeatedly find myself overawed by the staggering insignificance of what I know when weighed against what I don't! It helps to be reminded that everyone else had to start somewhere, too. Thanks for being so supportive of my efforts!Keep up the good work!

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I am really loving the theme/design of your website.Do you ever run into any browser compatibility issues?A few of my blog visitors have complained about my site not working correctly in Explorer but looks great in Chrome. Do you have any tips to help fix this problem?

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I'm not sure where you are getting your information, but good topic. I needs to spend some time learning much more or understanding more.Thanks for excellent info I was looking for this info for my mission.

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It's difficult to find experienced people in this particular subject, however, you seem like you know what you're talking about! Thanks

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Howdy! I know this is kinda off topic but I was wondering which blog platform are you using for this site?I'm getting fed up of WordPress because I've had problems with hackers and I'm looking at alternatives for another platform. I would be fantastic if you could point me in the direction of a good platform.

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What's up to every one, the contents existing at this web site are truly amazing for people experience, well, keep up the nice work fellows.

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