# How do decompose the skill “knowing linear algebra”?

According to some guys who give advice on how to master different skills, i tried to decompose what it means for me "to know about linear algebra". I tried to be very specific, to make my goals measureable to stay motivated and assure i get into the right direction with my effors. The last 2 times i read books from A to Z.

I wasnt very pleased with the results, as i forgot about the concepts really fast and wasnt able to transfer the learned stuff to other fields, e.g. optmization, machine learning or circuit design.

So here is what i came up with what i think it means when i know about linear algebra:

1) Know the important definitions and be able to write them down, exact and from memory.

2) For every definition or abstract concept, be able to deliver an example for a useful application.

However, after 1 month of putting in 40 mins to 1 h a day constantly i struggle to answer questions were i need to make the transfer or application of the definitions/concepts by myself. For some concepts where the books dont provide an example of application, i struggle to figure one out by myself. If the context of the question or sometimes even the notation changes, i feel lost. Do you have an advice on what exact aspects of learning i should focus or what techniques i could use?

It seems to me that your primary measures of success are:

• ability to remember and intuitively understand concepts
• ability to apply concepts in different settings (e.g. optimization, ML, and circuit design)

To that end, it doesn't seem to me that your idea of "what it means to know linear algebra" aligns well with your metrics of success. For instance, I don't think that "know the important definitions and be able to write them down, exact and from memory" is a particularly useful goal.

For example, I have never memorized the axioms of a vector space, but this has never stopped me from being comfortable with the abstract notion of a vector space. Instead of memorizing the 8 axioms, I remembered why it is we care about these 8 axioms. In particular, I would say that a vector space is any mathematical framework with a sensible notion of addition and scalar-multiplication, and the 8 axioms are there to clarify what exactly a "sensible notion" is here.

I also don't think it's necessary to have an application in mind for every single abstract concept. Rather, I think it's important to have some (any) notion of "why we care" about a particular concept. This could be an application, but it could also just be a description of the idea in terms of other things you understand. For instance, I couldn't tell you about a quick and easy application of matrix rank, but I can tell you that the rank of a matrix measures the dimension of the image of the associated linear transformation.

With that in mind, here's what I suggest: instead of trying to memorize concepts, try to apply them. Rather than reading textbooks cover to cover, go to the exercises at the end of each section and try to answer them; maybe even ask about those questions on this site, if you get stuck. If you manage to reach the point where you can answer both the very concrete, matrix-based questions like those in Strang's Introduction, and the abstract vector space questions like those in Linear Algebra by Friedberg, Insel, Spence and LADR by Axler, then you'll have a very robust and flexible internal model for the ideas of linear algebra.

Another thought is to get a better handle on how the different perspectives on linear algebra are connected. For instance, I think you will be in better shape if you make yourself comfortable with the fundamentals of linear algebra as they're presented in this series from 3blue1brown.

I hope you find this helpful.

• Thank you very much. I agree with you that my measures of success are more like the two points you mentioned. The reason why i had the definitions on my list are the proofs. To me it seems you can only proof things of you work with the definitions and apply logic reasoning together with them. However, prooving is not my goal, indeed. Can you eloborate how you think one can obtain the helpful answers to the why's you mentioned? From applying the concepts? – JohnnyS Dec 12 '17 at 9:02
• I think “applying the concepts” is the best way. As I said, I think your next step should be try some textbook exercises. More than anything, it’s solving problems which has taught me the importance of certain ideas and how they all fit together. – Omnomnomnom Dec 12 '17 at 14:10