
Linear Algebra by Jim Hefferon – free textbook - n-izem
http://joshua.smcvt.edu/linearalgebra/
======
nafizh
If you want to learn linear algebra by coding in python, this is hands down
the best book out there.

Coding the Matrix - Philip Klein [0]

It used to have a Coursera course, but I think it's been taken down. The
website has videos of the course taught at Brown I think.

The associated website is:
[http://codingthematrix.com](http://codingthematrix.com)

[0] [https://www.amazon.com/Coding-Matrix-Algebra-Applications-
Co...](https://www.amazon.com/Coding-Matrix-Algebra-Applications-
Computer/dp/0615880991/)

~~~
vasili111
The course is available here:
[http://academictorrents.com/details/54cd86f3038dfd446b037891...](http://academictorrents.com/details/54cd86f3038dfd446b037891406ba4e0b1200d5a)

It is legal, not a pirated version.

~~~
nelsots
That was a great course! I'm glad it's still accessible. I was not aware of
this resource for courses. There was a fantastic course on probability from
the University of Pennsylvania that disappeared when Coursera went through a
change a few years ago. Maybe I can find that one there too. Thanks for the
link.

~~~
nafizh
Did you find the Upenn course on probability? I have been looking for it for a
long time. The professor was absolutely brilliant.

------
jimhefferon
How nice to see this link here! I could answer any questions that anyone has.
--Jim

~~~
cs702
First, let me say THANK YOU for making this a free textbook available online
to everyone everywhere. The world owes you a debt of gratitude. I, for one,
will do my part and make a donation via the link at the bottom of your
textbook's web page. I hope others here will join me.

My question is: What motivated you to create this freely available book?

~~~
jimhefferon
Thank you for the donation. I use the funds to work on revisions of this text,
and on developing others.

For motivation: briefly, I was at Union College and was asked to teach Linear.
I saw a second edition of Strang's book, thought it was wonderful (still do),
and adopted it. But the students, who were perfectly good students, did not
get it. They just couldn't solve the problems that a person should solve in
that class. I decided after some reflection that the folks I had need to be
brought along to where they are ready for a higher level understanding. So I
put together a presentation that uses lots of examples, motivation, and
naturalness to try and develop maturity in people who do not yet have it.

(Again, I'm not talking Strang's book down, it is very fine indeed. But the
students that I have in front of me at this point are just not yet ready.)

------
nsajko
Ten years old discussion:
[https://news.ycombinator.com/item?id=351334](https://news.ycombinator.com/item?id=351334)

~~~
killjoywashere
If a link is getting hits with comments 10 years later, should there be some
sort of special class of points? Everybody else is running JV sprints and this
guy is dropping olympic marathons.

------
gballan
Here is another [1]--perhaps not to everyone's taste, but it was influential
for me. In particular, the first words of the preface: "The underlying spirit
of this treatment of the theory of matrices is that of a concept and its
representation."

[1]
[https://archive.org/details/LinearAlgebraAndMatrixTheory](https://archive.org/details/LinearAlgebraAndMatrixTheory)

~~~
thanatropism
My own take is that the "theory of matrices" (not a standard usage but should
be) is worth learning for some (sp if you're going into hardcore numerics like
stability of algorithms) but not for everyone. It's like studying calculus as
the theory of smooth curves -- it robs you of some potential for growth.

Also: some people just want "math for deep learning" and that's mostly baby
calculus: the chain rule and gradient descent.

------
release_cycle
What is nice about this book is it includes solutions to exercises. I wish
more proof-theoretic books included exercises in the back of the book like
this one does. Yes, you can prove something in multiple ways, and no that
isn't a good excuse to not include at least one version of the proof in the
back of the book. It really benefits the self-learner. The math community
seems to have a lot of artificial gatekeeping to keep non-academics out.

They do this in several ways.

1) They recommend textbooks to beginners that are too advanced for their skill
level.

2) These textbooks do not include solutions and the advice given is that
solutions would somehow rob them of the experience (don't let those that lack
self-discipline ruin learning for non-traditional students that aren't in a
formal classroom with access to professors and TAs). The self-learner needs
some sort of feedback system.

3) They tell the self-learner that solutions aren't provided because everyone
has the ability to know if their solutions are correct without having their
work checked. Would you write a complicated program and write no test cases,
or could you instantly know your program is bug free the first time you write
it? Why is math suddenly any different?

Despite popular elitist opinions, I'd recommend this book over Axler for the
beginner that doesn't know linear algebra. Everyone says Axler is the perfect
first book in linear algebra. It really isn't. Even Axler admits this himself
in the preface. He assumes that it is a second approach to the field.

But people like to be elitist and recommend books to beginners that aren't
always best from a pedagogical perspective.

Those are the same class of people that recommend Rudin or Spivak to someone
that wants to study elementary calculus 1 material.

~~~
skh
In my experience (math teacher at a community college for 20 years and former
Ph.D. student who quit before finishing the thesis) the overwhelming majority
of people can not learn mathematics properly from a textbook. They need a
teacher or someone to help with problems. It's very difficult to get to a
point of really understanding the definition/motivation behind concepts. Even
with a teacher it's hard to understand.

Here's a simple example. The distributive property says:

a (b + c) = ab + ac

I can teach students to expand 3(x + 2). With some practice almost all of them
will get this. They'll say

3(x+2) = 3x + 3(2) = 3x + 6

Do you know how hard it is to convince people that due to the nature of
equality you can reverse the steps? Some never understand that

3x + 6 = 3(x + 2)

Some will never understand that 3x + 2x = 5x because of the distributive
property and that combining like terms for the expression 3x + ax is the same
process.

I don't think there is any desire to be elitist in terms of having good books
that are appropriate. It's just hard to do. We've gone through various cycles
of reforming calculus, introducing old concepts in a new way. Through all of
the changes one thing has remained constant in my experience. The percentage
of people who can get it remains the same.

I do agree with your point on providing solutions.

~~~
roenxi
> the overwhelming majority of people can not learn mathematics properly from
> a textbook.

While you are right, a related issue is that most maths textbooks are
atrocious at most aspects that aren't writing pages of equations. There is
almost nobody sitting at the 3-way intersection of great mathematician, great
writer and great educator who can then write great textbooks.

I've found books with terms like "History" and "Philosophy" in the title are
much better places to learn about mathmatics; combined with wikipedia for
formulas and details. Any book with excercises but no solutions or historical
context has turned out to be basically useless to me, even as a reference
(wikipedia is usually better for simple stuff).

I've got something like 4 books on statistics on my shelf at the moment. The
only one that I've actually manged to read and learn something from has been
Chatterjee's Philosophy of Statistics, because it talks about what techniques
were developed in context of which problem, failed alternative approaches,
explains what was confusing to some of the greatest minds in the history of
statistics, etc. This has been vastly more useful in setting up a framework
for what the world of statistics looks like that I can attach a whole bunch of
proofs and suchlike too. It has been enlightening in a way that textbooks
can't really manage.

~~~
sriram_malhar
Did you mean "Statistical Thought" by Chatterjee?

~~~
roenxi
Oh, sorry, yes I did. "Statistical Thought: A Perspective and History". The
first half is basically all the schools of thought on how probability theory
and the real world interact; so I keep thinking of it as a philosophy book.

------
MrMorden
The lab on the cover of the lab manual deserves all the upvotes.

~~~
jimhefferon
It is kind of you to notice her. We had to say goodbye last year. She was a
sweetheart.

------
Mugwort
It's nice to have a free book as a supplement. Linear Algebra Done Right by
Sheldon Axler is a much better choice and when making an investment in
learning, particularly a difficult subject like linear algebra, saving money
with a free text is probably the most expensive thing you can do. You might
save $40 but end up not really learning and wasting hours you don't have.
Above all, find a good teacher.

~~~
romwell
Axler's Linear Algebra Done Right has always been my favorite linear algebra
text (I say this 10+ years after reading it and finishing my math PhD, if that
matters).

For those who want a free alternative, behold: Treil's Linear Algebra Done
Wrong[2]

The books is downloadable as a free PDF. The name is an answer to Axler's book
(dry mathematician's humor), and offers an opposite approach (getting to
determinants first).

While I agree with Axler and diagree with Treil, LADW offers way more examples
and applications, and _together_ LADR and LADW offer a complete, excellent
course material.

\---------------------------------------------------------

As for the linked text: not a bad text, but I wouldn't pick it over LADR +
LADW.

Here's why:

1\. size: it's larger than LADR+LADW taken together. It's hard to see the
forest behind the trees.

2\. exposition: it follows the structure of many other texts that I don't like
because they terribly confuse the students (that I'd have to re-teach
afterwards): starting with solving systems of linear equations, then jumping
into vector spaces, for example.

3\. I don't like how key concepts (matrix product, determinant are
introduced). If you already know the material, it will be hard to see what's
wrong with the approach of throwing a definition at the reader, and _then_
talking about why that definition was made. But the opposite should be the
case.

After teaching Linear Algebra, here's my litmus test for a _good_ book. At a
glance, it should make the following clear first and foremost:

1\. A _matrix_ of a linear map F is simply writing down the image of the
standard basis F(e_1), F(e_2), ... F(e_n). These vectors are the columns of
the matrix. If you know them, you can compute F(v) for any v by linearity.
That's called "multiplying a vector by matrix"; we write Mv = F(v).

2\. The _product_ of matrices is simply the matrix of composition of linear
maps that they represent. The student can figure out what that matrix _should_
be (or should be able to do so); here's how. If M is the matrix of F, and N is
the matrix of G (where F and G are linear maps), then the first column of MN
is F(G(e_1)) = M x (first column of N). Same for other columns. Ta-dah.

3\. The _determinant_ of v_1, .. v_n is simply the _volume_ of the lopsided
box formed by these vectors (mathematicians call the box "parallelepiped"). In
particular, in a plane, the area of the triangle formed by vectors A and B is
half the determinant. This are can have a minus sign; switching any pair of
vectors flips the sign.

4\. _Eigenvectors_ and _eigenvalues_ are fancy words that allow us to describe
linear maps like this: "Stretch this picture along _these directions_ by _this
much_ ". Directions are eigenvectors, by how much - eigenvalues.

Bonus:

5\. Rotation and scaling are linear maps. That's all _any_ linear map does:
rotates and stretches. Writing a map down in this way is called _singular
value decomposition_.

6\. Shears are linear maps that don't change the volume. Any box can be made
rectangular by applying a bunch of shears to it. That's called _Gaussian
elimination_ or _row reduction_ when you look at what happens to matrices (and
apply scaling as the last step). This is also an explanation of _why_ the
determinant gives volume (if you define it as an alternating n-linear form).

That's the beginning of a solid understanding of the subject.

From my experience, LADR+LADW leave the student with an understanding of 1-4,
and other texts, due to being organized badly, don't (even when they contain
all the information in some order).

\---------------------------------------------------------

Books I recommend:

[1]LADR: [http://linear.axler.net/](http://linear.axler.net/)

[2]LADW:
[https://www.math.brown.edu/~treil/papers/LADW/LADW.html](https://www.math.brown.edu/~treil/papers/LADW/LADW.html)

~~~
enriquto
> 6\. Shears are linear maps that don't change the volume.

Just to nitpick, but this sentence might be read ambiguously by some people,
who would understand it as the definition of shear. You may want to say
"Shears are an example of ...".

~~~
romwell
Thanks; I'm past the edit time cut-off, but will keep it in mind for the next
time I say something like that :)

------
nagVenkat
Are there any books like this for probability?

~~~
jsinai
First chapter of this book has a similar style and will get you started:
[http://mbmlbook.com](http://mbmlbook.com)

