Remembering Definitions and Theorems

25 Jul 2024

Whenever recalling a definition or theorem, I never look it up first.

Instead, I always:

  1. Write down whatever I can remember purely from memory.

  2. Polish what I have until it resembles a high-quality definition or theorem statement.

  3. Once I’m satisfied, I consult high-quality resources to verify and, if necessary, further refine the definition or theorem.

Points 1-2 relate to my recent discussion about internalizing mathematical concepts whenever possible.

I find it difficult to remember definitions or theorems verbatim, so I would not normally recommend this.

Instead, use your experience in dealing with the types of problems that use the definition/theorem to reconstruct a suitable definition/theorem statement.

Let’s pick one example: “Give the definition of an invertible matrix.”

My thought process would be something like the following:

Right, invertible matrices. Okay, let’s define a matrix $A.$ Only square matrices are invertible, so let $A$ be an $n \times n$ matrix. Now, what does it mean for $A$ to be invertible? Well, it means there exists another $n \times n$ matrix, call it $B$, such that when I multiply the two together (in either order), I get the $n \times n$ identity matrix.

Huh, it seems like I have it. Okay, let’s try to write a proper definition:

Let $A$ be an $n \times n$ matrix. The matrix $A$ is invertible if there exists an $n \times n$ matrix $B$ such that $AB = BA = I_n,$ where $I_n$ is the $n \times n$ identity matrix.

I haven’t looked up this definition in years. But after writing the above, I checked a few resources, and this definition looks fine. I could have used slightly different phrasing or notation, but the essence is correct, and all important details are there.

Once you’ve practiced this process a few times (using spaced repetition, of course), you’ll find that you can write down the definitions more accurately and quickly.