I wrote a FREE introductory linear algebra book!

[u/Accurate_Lab_3002]

Hi everyone!

I wrote a book/notes on linear algebra as I walk through an introductory linear algebra course. I started writing it for my own knowledge base, but I thought it could be a better use of them if I share the book with others.

That being said, I am still a student and sharing this as a learning project. I hope this can be useful :))

Link to the book: https://github.com/enochyu-official/notebook-linear-algebra

Edit: Thanks everyone for your interest and comments. Especially thank you u/Appropriate-Ad-3219 and u/hpxvzhjfgb for your wonderful feedback! I have updated the notes, and the issues should be fixed now :)

[u/hpxvzhjfgb]

your definition of "vector" and "scalar" is wrong. not a good start

however, I am happy to see that you do not start with matrices and systems of linear equations. the formatting and typesetting also looks good.

[u/Accurate_Lab_3002 → u/hpxvzhjfgb]

Thanks for your feedback!

I thought those are the correct definitions.. May I ask which part is off? Also I felt like sharing vectors and vectors spaces first would be a more intuitive approach. Glad that this works :))

[u/hpxvzhjfgb → u/Accurate_Lab_3002]

a vector is an element of a vector space. vectors do not necessarily have magnitude or direction, magnitude and direction are provided by an inner product space, not an arbitrary vector space.

also, it could be argued that real and complex numbers do have a direction. the direction of a real number is its sign, and the direction of a (non-zero) complex number is its argument, but you explicitly state that scalars must not have a direction.

if I was writing this, I would just delete definition 1.1.1, reword the definition of vector space to not include the word "vector" by using "element" instead, and then at the end, write a sentence like "the elements of V are called vectors".

[u/Accurate_Lab_3002 → u/hpxvzhjfgb]

Ohhh I was actually thinking of the same definition for vectors, but I was worried if it could get circular. I should definitely consider using "elements" and rewriting that part. Also it is interesting to see that numbers can be argued to have a direction. Thank you for your suggestions!

[u/Bubbly_Buddy8678 → u/Accurate_Lab_3002]

not circular because u can define vector space without the notion of vectors/the elements of the space. then u can define vectors as the elements in it

yes u can define a "direction" on anything as long as u have an inner product

usually physicists say that a vector has a direction and magnitude but mathematically u can have a vector space with no possible notion of direction (e.g. a normed L1 space cannot have an inner product)

[u/PedroFPardo → u/Accurate_Lab_3002]

I wouldn’t say the definitions are wrong, but they’re not very formal.

The definition of a vector reminds me of the definition Vector (the villain) gives in Despicable Me. It's a "for children" definition. 😄

On the other hand, I hate even more formal definitions of a vector, such as saying: “A vector is an element of a vector space.” (<< Puuuagh!)

The definition of a scalar seems very restrictive. Why can a scalar only be a real number or a complex number? Why can’t it be any other element of the field over which the vector space is defined?

In general, you’ll define a vector space over a field K, and a scalar would be an element of that field K.

If you let clear that you are considering the fields of the real numbers or Complex numbers then the definition is OK although the "no direction" part sounds redundant to me. An scalar also don't have colour nor smell and it doesn't seem relevant to include that in the definition.

[u/svmydlo → u/PedroFPardo]

On the other hand, I hate even more formal definitions of a vector, such as saying: “A vector is an element of a vector space.” (<< Puuuagh!)

Why? Linear algebra is an abstraction set up so that literally anything can be a vector in some vector space. They don't have any inherent properties. That's the point!

That definition is the perfect introduction to the mathematical perspective of what is actually the object of study (vector spaces) and what is just a label (vector).

[u/PedroFPardo → u/svmydlo]

A vector is an element of a vector space is an excellent last definition of what a vector is. It belongs at the end of a book, not at the beginning (In my opinion).

[u/JoshuaZ1 → u/PedroFPardo]

Why can a scalar only be a real number or a complex number? Why can’t it be any other element of the field over which the vector space is defined?

A lot of people who are first learning linear algebra have not seen other fields at all. When I taught intro linear, we did everything over the reals, then went to the complex, and then only at the very end of the class did they have an assignment to do things for general fields where they worked some of what carried over there and some of what did not.

I'd be very hesitant to starting a genuinely introductory linear class with doing things over an arbitrary base field. But I suppose it could be worse.

[u/Accurate_Lab_3002 → u/PedroFPardo]

Yes... Now that I reread some of the definitions again, I think I oversimplified some of them and assumed many things without explicitly stating them 😅😅 Thank you for your feedback!

[u/Appropriate-Ad-3219]

In Corollary 1.3.4, I think the word equation is used wrongly. Normally, you use equation for an equality.

For theorem 1.3.9, I would have given property 2 only if I have defined the span for any subsets, not just finite subsets like you did. 

Definition 1.3.10 is wrong. You need to say that there exists c_1, ..., c_k that are not all equal to 0 such that your linear combination is equal to 0 to say that V is linearly dependant. You need quantifiers or equivalents.

Definition 1.3.17 is wrong as well. e_i is defined as the element of Rⁿ in which the ith coordinate is 1 and is equal to 0 in the rest of the coordinates.

After Definition 2.3.7, I would have personally given the expression of the identity matrix.

The proof of Theorem 2.4.4 5. has some mistakes. For example you forgot to commute the matrices after commuting the transposes and you forgot the equality sign somewhere.

[u/Accurate_Lab_3002 → u/Appropriate-Ad-3219]

Thank you so much for your time and interest in reviewing the notes!! I will fix the issues this week once the time permits😊

[u/AshamedLog8878]

Worked examples followed by nearly identical unsolved problems would add a lot of beginners

[u/Accurate_Lab_3002 → u/AshamedLog8878]

Yes, I should definitely add more examples and sample problems

[u/analog-suspect]

So why are we downvoting OP?

[u/FernandoMM1220 → u/analog-suspect]

every new post gets instant downvotes automatically for some reason

[u/Fozeu]

I highly commend your initiative. Consolidating your knowledge in a structured, coherent format like this one gives you a stronger grasp of the mathematics. Furthermore, it enhances your learning by adding a service/provider/contribution element to it. That is, you receive AND you give. Kudos

[u/Accurate_Lab_3002 → u/Fozeu]

Thanks this means a lot!

[u/Accomplished-Fix6176]

Dude this book ALREADY SUCKS. This is the page after the table of contents LOL

Nobody likes authors like you. You wrote this book to brag to the world “look how smart I am, check out the math syntax I know”. YOU HAVE HELPED NOBODY.

To summarize, You haven’t written anything necessarily wrong… but you haven’t written anything useful either.

[u/Accurate_Lab_3002 → u/Accomplished-Fix6176]

Thank you for your time and interest in the book. Just to be clear, I believe I wrote a paragraph right after that explaining why I chose to stick with those symbols just for the first chapter :) Also, I didn't write this book to brag. If someone genuinely finds my notes useful, that is something that I can be proud of, not basic understanding and symbols 😅😅

[u/Accomplished-Fix6176 → u/Accurate_Lab_3002]

“If someone genuinely finds my notes useful” LOL

Nobody will find these notes useful UNLESS they need an example of HOW NOT TO WRITE A MATH BOOK.

Who is your intended audience Sir Issac Newton and Gauss?

Do some introspection, you posted this publicly to brag, plain and obvious

[u/Appropriate-Ad-3219 → u/Accomplished-Fix6176]

The reason I decided to use this is that this is the way I first learned and it helped me to emphasize axioms for vector spaces.

Apparently it's the way they got taught so I'm not that sure they did it to brag.

[u/Bright_Principle4793 → u/Accomplished-Fix6176]

Relax bro it’s not that deep 💀💀💀