What kind of requests would you like to get? Specific questions? Certain subjects? etc.
The more specific the better, basically.
One thing missing: what is the intuitive “geometric” interpretation of matrix transpose? Considering the transpose is used all the time, I still don’t have an intuitive sense of what it does.
It’s unclear to me what counts as a “geometric” interpretation here. The coordinate-free interpretation is that taking transposes is the coordinate-dependent version of taking the adjoint of a linear operator between inner product spaces. There’s also an interpretation in terms of SVD which may be more explicitly geometric: taking transposes swaps the left and right singular vectors. (This is one way to think about the spectral theorem: it means a symmetric matrix is a matrix whose left and right singular vectors agree, which means they’re eigenvectors. This isn’t quite a proof because singular vectors aren’t quite unique but it’s pretty close.)
An intuitive explanation of the SVD.
I wrote a blog post about SVD; not sure whether it will give you what you want but I like it.
an explanation of the difference between transfinite induction and regular induction. I.e. IIRC, there needs to be defined a whole new kind of induction in order to do transfinite induction vs the base induction used to define things a little past N. Why is another mechanism necessary?
As gjm says, ordinary induction is a special case of transfinite induction so it’s not exactly a whole new kind of induction, just a (pretty mild, all things considered) generalization.
“Geometric” intuition is basically the way that the 3Blue1Brown YouTube channel would explain things. I’m not sure if you’re aware of it, but their “Essence of Linear Algebra” goes through the broad high-level concepts of linear algebra and explains them, with a very visual/geometric intuition for things like basis change, inverses, determinants, etc.
The more specific the better, basically.
It’s unclear to me what counts as a “geometric” interpretation here. The coordinate-free interpretation is that taking transposes is the coordinate-dependent version of taking the adjoint of a linear operator between inner product spaces. There’s also an interpretation in terms of SVD which may be more explicitly geometric: taking transposes swaps the left and right singular vectors. (This is one way to think about the spectral theorem: it means a symmetric matrix is a matrix whose left and right singular vectors agree, which means they’re eigenvectors. This isn’t quite a proof because singular vectors aren’t quite unique but it’s pretty close.)
I wrote a blog post about SVD; not sure whether it will give you what you want but I like it.
As gjm says, ordinary induction is a special case of transfinite induction so it’s not exactly a whole new kind of induction, just a (pretty mild, all things considered) generalization.
“Geometric” intuition is basically the way that the 3Blue1Brown YouTube channel would explain things. I’m not sure if you’re aware of it, but their “Essence of Linear Algebra” goes through the broad high-level concepts of linear algebra and explains them, with a very visual/geometric intuition for things like basis change, inverses, determinants, etc.
Unfortunately, they never covered transpose :)
Also, I’ll take a look at your blog post, thanks!