Figuring Out Math Concepts with Ask Gemini

I've always wanted to learn more math.

I have made efforts here and there, but the overwhelming challenge, and my lack of time due to effort intensive work such as at Hexmos has kept my overall progress limited.

These days, with LLMs, I have a renewed hope that -- perhaps, even a time poor person like me could get a decent grasp of mathematical ideas -- with a fraction of time investment compared to what it used to take before.

My goal here is "Exposure", as Sean McClure tends to call it, "exposure":

Anyway, how does one go about familiarize with Mathematical concepts with the help of LLMs?

Alright, let's get started with something practical.

I remember studying "eigen vectors" and such long back in my college days, and I'd like to get a refresher on it.

And I already know that I like going through problem/solution sets or rather worked out examples, rather than going through tons of theory first.

I am a "problem-first" explorer of topics.

So I also happen to know that "Schaum's outlines" are good for this sort of stuff.

And I google "Schaum's linear algebra" and get this

The first link seems to have a PDF of some sort, let's check it out:

Then, in the text, I search for "Eigen", and I get the following. Also I hit "Ask Gemini" on the sidebar, and ask it a basic question to check whether it has context on the presently active page (page 299) -- it does!

What catches my attention now is -- "two points of view", and that catches my interest.

So I open a skill in Gemini "Break it down", and ask it "What are these two points of views"

Gemini has these skills and one can add many other useful skills (idea - perhaps I can get it to quiz me to check my understanding -- something to check later!)

A good start here -- the perspectives are "formula and numbers" vs "geometry and spaces"

OK -- noticed there are quizzes and flashcards too, pretty cool:

I'm reading this second paragraph, and I happen not to immediately have in my memory what is "diagonalization" or what is "invertible matrix".

I think dealing with "unknown terms" is a key operation in any kind of mathematical activity.

Now, I can proceed in two ways:

1. Ask Gemini to find the relevant part of the text
2. Go the Ctrl-F route I went earlier.

Also, as a side thought, I am starting to realize, reading a mathematical text with AI requires these sorts of "bookmarks" to switch between a session for various concepts.

Having a free-flowing chat is fine, but then moving between references of the book can be way more powerful.

Perhaps something like Google's NotebookLM can do a better job at this, but now, we will make do with Ask Gemini, and see if it can handle a question about finding content in the book itself.

I also check whether the sidebar can take actions on my behalf (scroll to the page where concept x is introduced)

Gemini responds -- gives me the section number, but it cannot scroll for me (how disappointing!). The blue link you see just links to the PDF top - no page number reference. This makes me think "Ask Gemini" doesn't have page number context at all or in a usable way.

But the information is legit, I was able to Ctrl-F the piece of text, and find exactly what Gemini promised me:

Gemini gave an answer of its own along with the previous response, but I found a better example in the book itself -- essentially multiply A x B, or B x A, you get a singular matrix -- and here A and B are inverses:

Now I am ready to go back to diagonalization, I wish the AI could help me navigate across pages as I need them, but there is no such facility as of now. So I find my own way with Ctrl-F for now.

I wanted an ELI5 skill for this "Ask Gemini", but all I see is a "simplify" one:

The whole skill is available for inspection:

Objective: Transform complex content into a simple explanation a 5 year old could understand

1. **Identify the central point of the content**

    * Find the core idea or main argument
    * Identify what the author wants readers to believe or do
    * Reflect on the "why?" of the content
    * Note the scope and limitations of the content

2. **Assess complexity**

    * Identify technical terms, jargon, or specialized vocabulary
    * Note assumed knowledge or prerequisites

3. **Make abstract concepts concrete with analogies**

    * Connect to everyday experiences (for example, playground, kitchen, family)
    * Use physical objects to represent abstract ideas
    * Create simple cause-and-effect relationships

4. **Rewrite using language accessible to a kindergarten student**

    * Replace jargon with common words
    * Shorten sentences to one idea each
    * Use active voice and present tense
    * Avoid hedging words like "might", "perhaps", and "arguably"

5. **Handling exceptions**

    Prioritize excellent content in your response. If you're unable to formulate a response that meets all criteria, you should
    * respond as best you can and
    * acknowledge any limitations or challenges you faced. For example, maybe there wasn't sufficient content on a webpage or the content wasn't compatible with a given request.

    Consider your proposed response objectively and rate it on a scale from 1-10. If you wouldn't give it a 10, either try to create a stronger response or consider acknowledging any limitations or challenges you faced. The score is just for your own purposes; don't share it with the user.

6. **Final response**

    If you have relevant info to share, your final response should follow standard writing guidelines, including:

    * Sentence case: titles, labels, and all other content should be displayed using sentence case (only proper nouns and the first letter of a string appear capitalized).
    * Favor simple sentences that use common words

    Your final response should also be:
    * 3-5 short paragraphs
    * Under 200 words total
    * No jargon or technical terms
    * At least one concrete analogy

7. **Follow-up questions**

    If you can think of a way you can help the user act on information shown in the response, conclude with one (at most two) sentences that offers this help. Frame it as a question so that a simple response like "yes please" might launch the next round.

Let's trigger an explanation with this:

This is the explanation I got, and it doesn't make me feel better. It seems to be giving some toy analogy which made it worse (for me). It's like you can either spin the tray to arrange toys or you can look at it from another perspective (!). The robot teacher is leading me astray here.

Need something better. Let me try a hand-rolled prompt:

Alright, much better - because it combines both technical analogy and ELI5 analogy:

I am starting to get an intuitive feel for what the whole thing is about. The matrix view is not very meaningful to me. I can imagine points in space. I can also imagine them being "crooked" or "messy". Then we change our default point of view, using change of basis. I know what basis vectors are from history, so I have a sense of what these are. Basis vectors are the "core direction givers" for any object we put in a vector space. With these basis vectors and a multiplier matrix or such, we can regenerate the object. So these basis vectors are "compression objects" of sorts. Eigen vectors are something of the sort.

I like the ELI5 example much better here -- it actually helps.

You have a crooked paper hanging in front, it's awkward to draw on. You want a straight line (90 degree like). So you change your chair position so that from your perspective it is crooked anymore, then draw, then put the chair back in position. This explains it best:

The formula $B = P^{-1}AP$ says: Change your perspective ($P$), do the action ($A$), then switch back ($P^{-1}$). The final result $B$ is the exact same action, but expressed in the easiest way possible!

Now I have an intuitive sense of what is going on here. Now let's see where this is used most in terms of applications.

PageRank, PCA, phew -- this is everywhere, something truly important.

Concluding thoughts

Overall, this experiment left me fairly optimistic.

I don't think LLMs replace textbooks.

They make textbooks much easier to read.

A few things stood out to me:

• The textbook remained the source of truth. Gemini was most useful when explaining, connecting ideas, or helping me locate information.
• Unknown terms stopped being roadblocks. They became quick detours before returning to the main flow.
• Prompt quality mattered a lot. The built-in "Simplify" skill wasn't particularly helpful here. A hand-written prompt asking for both intuition and the technical explanation produced much better results.
• Applications made concepts stick. Once diagonalization clicked, seeing PageRank and PCA immediately answered the question: "Why should I care?"

There are still some missing pieces.

• Navigation is the biggest one. Gemini could tell me where something was, but it couldn't take me there.
• Page references would help enormously. "Jump to page 284" feels like an obvious capability.
• Bookmarks and multiple threads would be even better. Mathematical reading naturally branches into prerequisites and side quests. A chat interface isn't ideal for keeping track of all of them.

I'm curious to see where these tools go over the next year or two.

For now, though, this already feels like a much more enjoyable way to work through technical books than constantly bouncing between PDFs, Google searches, and Stack Overflow.