How can one think intuitively about (linear) algebra?

Question

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).

Answer 1

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.

Answer 2

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.

Answer 3

You ask some good questions - basically, you want to express things in a coordinate independent way. The following books

Linear Algebra Done Right by Axler and

Linear Algebra Done Wrong by Sergei Treil

both try to emphasize linear transformations point of view and that matrices arise when coordinates are chosen, although I don’t know if they answer For example SVD saying $A=U\Sigma V^T$ intuitively to me can be interpreted to that after rotation of domain and codomain, every linear transformation is scaling appropriate axis by the singular values. Orthogonal diagonalization of symmetric matrix has a similar interpretation.