[anateus]

I appreciate the visualizations, but the explanatory approaches follow the same ones I've seen in basically every textbook. The following particularly epitomizes that: "...it is worth defining what a subspace is in terms of its formal properties, then what it is in laymans terms, then the visual definition, showing why it is that those properties need to be satisfied."

Why not... show it visually, manipulate it visually, explain in layman's terms and THEN explain the formal properties? That is the approach 3Blue1Brown takes to great success.

So, definitely useful renderings, but I'm not sure it's any more "intuitive"* than the mainstream approaches.

* In quotes to indicate that there is quite a bit of subjectivity in that term.

[smspillaz]

Thanks for the feedback! I was a little worried that the explanations would fall back too much on formal properties, so I guess I've got a bit more work to do on the visualization side of things.

[mannykannot]

I think you are doing a great job here. I looked through your sections on determinants and inverses, and it is so much better than the way I was taught: "this is the formula for inverting a 2x2 matrix. This bit is called the determinant." For me, learning always works best when I know what we are working towards and seeing how it works, and putting the formal definition first is rarely effective in that. Start with a specific example, concrete where possible, and generalize from that.

[smspillaz]

Indeed. I think one of the thing I wanted to focus on as well in this series was explaining exactly where a lot of the formulas came from. It was a great learning experience, since a lot of what I was taught was basically as you said "here is the formula for the determinant and oh look, NxN determinants can be worked out in such and such a way if you just apply this algorithm". It was super useful to decompose why that algorithm works from a visual perspective. Same thing with surface integrals.

[fjsolwmv]

There's no perfect order for everyone. The trick is to spiral around revisiting levels and styles (and don't worry if you don't grok a part yet) until they all start to gel and your brain forms aweb of connections.

So many people think the second book they read on a topic is soooo much clearer than the first. But it doesn't much matter which book they dead first vs second :-)

[mannykannot]

The 'second book effect' is an interesting point, but that phenomenon, by itself, would suggest that there are improvements to be made in how we teach things from the beginning.

[sannee]

> explain in layman's terms and THEN explain the formal properties?

The explanations are definitely very lacking of formalisms. There even isn't any proper explanation of what a vector space actually is.

It doesn't seem to pretend to be a full textbook replacement, more like a supplementary material, so I don't think it matters all that much.