# How can one think intuitively about (linear) algebra?

This may sound like a philosophical question, but it’s intended as a very practical question.

Broadly, I have two ways of doing math:

1. blindly following definitions and shifting equations around.

2. having a deep understanding of the meaning of terms and why theorems hold, allowing me to see immediately why a conjecture must hold, or what the solution to a problem will look like. I may not even need to do any formal derivations, and if I do, they flow immedaitely from my intuitive understanding.

With respect to linear algebra (and abstract algebra, though this question focuses more on linear algebra), I am to a large extent in (1), and I want to get to (2).

Mainly due to watching a 3blue1brown series, I have an intuitive understanding of

• what an eigenvalue and eigenvector is geometrically
• that a matrix represents geometrically, a linear operation on e.g. a Euclidean vector space
• other basic stuff

But when it comes to other concepts, whenever I use them I am really just shifting equations around mindlessly:

• matrices can be decomposed into a diagonal matrix and two other matrices that are inverses of eachother
• what a quadratic form represents geometrically
• that symmetric positive definite matrices have symmetric square roots, but positive definite matrices don’t necessarily have symmetric square roots.
• that symmetric matrices have real eigenvalues The sum of eigenvalues of a matrix are equal to its trace, and the product of eigenvalues equal to the determinant.
• etc etc

Alot of these things I can prove by mindlessly sequencing equations like a computer-based theorem prover, but I cannot immediately see why they are or are not true.

What can I do >>practically<< to gain this type of intuition quickly about lots of these linear algebra questions?

I feel like just doing more theorems, and solving more problems, won’t help. Are there good books that treat these things intuitively? Or a collection of methods to use?

(Note that I am entirely self-taught in this and have no formal math degree).

• The usual remedy is to read a good textbook that explains the intuition behind these concepts. – Rahul May 17 '18 at 7:25
• I recommend Gilbert Strang's book Linear Algebra and its Applications. Before diving deeper into mathematics, Strang tries to give the reader an intuition about the concepts, such as linear dependence and the like. Moreover, you find many practical examples where you put the Algebra to use, such as network problems, for example. – YukiJ May 17 '18 at 8:25
• I recommend this: youtube.com/… – harshit54 May 17 '18 at 10:47
• @harshitjoshi. Thank you but ive already seen it. Its really good. (See my question) – user56834 May 17 '18 at 11:24
• You need to play with definitions, try to come up with statements you want to prove, and prove them. Definitions are usually really really distilled, that's why some of them don't seem intuitively obvious or "right" – Jo Be May 17 '18 at 11:26

## 2 Answers

Find a good textbook - perhaps even course notes - there's plenty that can give very good and detailed explanations of many things in linear algebra. I cannot think of any reference off the top of my head right now. However, I do want to say that you can gain a very good intuition for linear algebra just by doing linear algebra, or doing other things that require it. At least in my experience.

• "you can gain a very good intuition for linear algebra just by doing linear algebra, or doing other things that require it." I think that depends on what exactly you're doing. I've been doing statistics, and mainly the books just take theorems in linear algebra for granted. Other textbooks on applications also do this. On the other hand, more pure mathy textbooks often focus heavily on proofs without intuition. The optimal seems to be something in between, which connects the theorems to intuition. – user56834 May 17 '18 at 10:24
• I think you're right @Programmer2134. Perhaps I should clarify. I think most texts certainly do take many facts from linear algebra for granted. I guess what I meant is when you see linear algebra in the context of something else, be it geometry, representation theory, statistic and so on, then sometimes you can learn the linear algebra as you see it, or based on what's required. The upshot is I guess you also see how it applies in different contexts rather than pure linear algebra, which could perhaps provide some intuition. I do stress that doing things this way may certainly not be the best – NickB May 17 '18 at 13:35

Intuition can be understood in different ways, according to a kind of "first intuition" or "reasoning" ad hoc that leads to that understanding (on this point more than one philosopher has worked).

The word intuition can also be accompanied by adjectives reducing the general semantics of it such as, for example, "geometric intuition" (which was relegated to a second plane in Analysis with the Weirstrass curve that is continuous everywhere but not derivable at any point).

Reducing the issue to linear algebra, there are "counterintuitive" examples, such as linear functions that are not continuous at any point of which there is a "surprising" example with the "very familiar" derivative in the space of continuous functions defined in $[0,1]$ with the Supremum norm.

I think that the "almost perfect" intuition that you want to deal with, is nothing other than the consequence of a consolidated knowledge of the subject which is in turn a consequence of a prolonged and very attentive study of linear algebra.