Develop good understanding of Linear Algebra

Question

I am self studying Linear Algebra from book by Kenneth Hoffman and Ray Kunze, and currently I'm on 2nd Chapter of vector spaces, though the text is easy to follow, specially the exercises that follow at the end of each section, after going through the theory but still I found it a bit dry.

What I earlier thought before picking this book, that once I do all the theory and exercises I'll have a good understanding of Linear Algebra but now I can only solve the problems. And do not think that now I'm developing a good understanding of the subject.

The text seems too dry, What I basically have to do is go through loads of Theorem-proofs-corollary-lemmas followed by a dozen exercises.

I googled to find some supplement notes on this book, but couldn't find anything good.
So, my question is- what else can I do/read to develop good understanding of the concepts other than just reading from Hoffman -Kunze? I 've heard a lot about Linear algebra done right by Sheldon Axler, Is this book suitable to satisfy all those things? Kindly mention any lecture notes, video lectures or any other text book you guys know of.

Thanks in Advance.

Answer 1

Try Matrix Analysis and Applied Linear Algebra by Carl Meyer. I think the book explains things very nicely and has many practical applications.

Answer 2

I studied Linear Algebra as an undergrad. I struggled with understanding the material at the time. Until years later...

At first I wanted to fit my understanding of Linear Algebra into ideas I had already mastered. I didn't understand n-dimensional spaces because I kept trying to create a visual image based on Cartesian 3-space. This is an incorrect way to go about it. I figured it out how to properly view these n-dimensional spaces, through writing a complex computer application to dynamically generate static files. Though, imagination is involved so my visual picture is likely different than another persons.

Knowing the fundamentals: basis, span, linear independence, vector space, is important. If you go through these too quickly, you could fool yourself into thinking you understand the terms, when you really don't. ...I didn't know what basis was for a long time, it just didn't click. Eventually I figured it out through the study of...

Linear transformations are a critical impasse or crux of understanding for L.A. Once you start to understand the general form of a Linear transformation you will begin to uncover your understanding.

Finding a book where you understand the syntax is important. You need a balance. Too esoteric and you may fool yourself into thinking you understand a concept you actually don't, and too nitty-gritty and you may suffer from fatigue while trying to read through examples, or may becomes overly focused on some small detail and miss out on the 'big picture.'

The subject matter is tricky. I can't speak for others but I basically needed a breakthrough in understanding to finally get it. Copying definitions, and going through examples will only get you so far. That's why it is so hard to learn, because it's on you, as the student, to really get it.

Don't be ashamed if you don't understand it right away, it took me a long, long time to get it... and I'm fairly bright. It's a beautiful subject of Math once you start to understand. I can recommend the Howard Anton book "Elementary Linear Algebra", Gil Strang's first couple of MIT lectures (I've only watched the first few), and a book by I.M. Gel'fand called "Lectures on Linear Algebra" in addition to some online resources. If you perform ERO to convert a Matrix to RREF once, you've done it a thousand times. I'd skip the silliness with those examples and move into proofs right away and this sort of thing :-). Then onto Linear transformations from different vector spaces or sets of varying dimension, etc. Don't over-glorify the subject right away, be patient. Good luck.

Answer 3

There is a nice video series on youtube by 3blue1brown called "the essence of linear algebra." I like this, because he opts for a more geometric approach. I think the key is striking a balance between the "geometry" of linear algebra and the algebraic side of things. For example, a linear transformation that is idempotent ($P^2=P \circ P=P$) is also a projection. It's nice to try and bridge these concepts. The same goes for the determinant as a group homomorphism/ area function etc.

These kinds of exercises helped me at least.