4 ms·

His description of the determinant, too. I hadn't heard that explanation until a second semester of real analysis, when learning the proof of the Inverse Functi

His description of the determinant, too. I hadn't heard that explanation until a second semester of real analysis, when learning the proof of the Inverse Function Theorem (an amazing thing to study, by the way, connects many the dots between linear algebra and calculus). Even then, it was a question I had to ask my brilliant, constantly-pissed looking young professor. "Hey, uh, the Jacobian... what does the determinant mean, uh, geometrically?". He looked at me like a slug, before explaining it was the measure of the newly mapped unit square. Fireworks went off in my head. Two linear algebra classes before only ever explained it by its algorithm or its usefulness (e.g., ∃ A^(-1) for A \in R^{n,n} iff det(A) != 0) Side note: that professor had the most effective teaching style for pure math I've ever seen. Besides lectures that expanded on the contents of Rudin and interesting problem sets, he gave us a list of a hundred theorems, propositions, and exercises. Told us the final exam would be six problems, four of which would come from that list, another of which would be a clever new one, the last something truly hard. Never learned analysis better than when sitting down and working through (not memorizing) each of those proofs and theorems for possible later recapitulation.

wicker 14y ago
Any chance you know where I could find that list or something like it?
kens 14y ago
Determinant as volume of the transformation is cool, but I still don't "get" the determinant intuitively despite many years of math. In particular, why does the determinant work to solve linear equations (i.e. Cramer's rule)? And what's the motivation behind the formula for the determinant? (I realize these questions are a bit vague, but I'm hoping for a more intuitive answer than "that's just the way the math works out".)
- jules 14y ago
  One approach to this is geometric algebra. I couldn't find a good reference that explains it intuitively, but there's this: http://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra#Matrix_Related http://en.wikipedia.org/wiki/Comparison_of_vector_algebra_an...
- vjk2005 14y ago
  reg. "And what's the motivation behind the formula for the determinant?", if I'm understanding your question right, the neat little pic at http://en.wikipedia.org/wiki/Determinant#2-by-2_matrices http://en.wikipedia.org/wiki/Determinant#2-by-2_matrices - might be what you were looking for.
- adeelk 14y ago
  As is often the case, the way to get the intuition here is to just work out the computation. Once you do that, it should all "click".
- jimhefferon 14y ago
  > And what's the motivation behind the formula for the determinant? FWIW, that book has two explanations: the first on p 296 is a lot like "that's how it works out" and the second on p 320 is geometric. > why does the determinant work to solve linear equations (i.e. Cramer's rule)? Does the explanation on p 331 of joshua.smcvt.edu/linearalgebra/book.pdf help? (It uses the geometric understanding of the determinant.)
kylebrown 14y ago
I wish that I'd had a similar experience. I bought Halmos "Linear Algebra Problem Book" based on the accolades, but the lack of motivation/inspiration caught me offguard and left me sorely disappointed before the end of chapter 1. If you can appreciate maths without pictures, you have to be either truly gifted or totally deluded. Recently, learning to do Principal Components Analysis to solve a handwriting recognition problem is what finally shed an enchanted light on linear algebra. The way that a simple matrix of data samples is transformed into an ordered set of principle components (eigenvectors, ordered by eigenvalue) is.. "unreasonably effective" (as they say). The principal components are your signal, and the rest (with eigenvalues ~0) are the noise. the handwriting recognition, btw, works fantastic for my simple application. no need for non-linear kernels and whatnot.
- textminer 14y ago
  Did you see that on Jeremy Kun's great blog? His primers are how I recently got into building my own Entropy-trained decision tree class and also got an intuition for PCA as a reduced basis. I had used truncated SVD and Fourier bases many times before, but to see it with images (eigenfaces!) really sold the intuition. Even better now, in this hacking life after pure math in college and grad school, is that I can build intuition now not just by proofs and exercises, but also efficient, coded implementation. Gives a different feel for the tools and concepts.

wicker 14y ago

Any chance you know where I could find that list or something like it?

kens 14y ago

Determinant as volume of the transformation is cool, but I still don't "get" the determinant intuitively despite many years of math. In particular, why does the determinant work to solve linear equations (i.e. Cramer's rule)? And what's the motivation behind the formula for the determinant? (I realize these questions are a bit vague, but I'm hoping for a more intuitive answer than "that's just the way the math works out".)

jules 14y ago
One approach to this is geometric algebra. I couldn't find a good reference that explains it intuitively, but there's this: http://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra#Matrix_Related http://en.wikipedia.org/wiki/Comparison_of_vector_algebra_an...
vjk2005 14y ago
reg. "And what's the motivation behind the formula for the determinant?", if I'm understanding your question right, the neat little pic at http://en.wikipedia.org/wiki/Determinant#2-by-2_matrices http://en.wikipedia.org/wiki/Determinant#2-by-2_matrices - might be what you were looking for.
adeelk 14y ago
As is often the case, the way to get the intuition here is to just work out the computation. Once you do that, it should all "click".
jimhefferon 14y ago
> And what's the motivation behind the formula for the determinant? FWIW, that book has two explanations: the first on p 296 is a lot like "that's how it works out" and the second on p 320 is geometric. > why does the determinant work to solve linear equations (i.e. Cramer's rule)? Does the explanation on p 331 of joshua.smcvt.edu/linearalgebra/book.pdf help? (It uses the geometric understanding of the determinant.)

kylebrown 14y ago

I wish that I'd had a similar experience. I bought Halmos "Linear Algebra Problem Book" based on the accolades, but the lack of motivation/inspiration caught me offguard and left me sorely disappointed before the end of chapter 1. If you can appreciate maths without pictures, you have to be either truly gifted or totally deluded. Recently, learning to do Principal Components Analysis to solve a handwriting recognition problem is what finally shed an enchanted light on linear algebra. The way that a simple matrix of data samples is transformed into an ordered set of principle components (eigenvectors, ordered by eigenvalue) is.. "unreasonably effective" (as they say). The principal components are your signal, and the rest (with eigenvalues ~0) are the noise. the handwriting recognition, btw, works fantastic for my simple application. no need for non-linear kernels and whatnot.

textminer 14y ago
Did you see that on Jeremy Kun's great blog? His primers are how I recently got into building my own Entropy-trained decision tree class and also got an intuition for PCA as a reduced basis. I had used truncated SVD and Fourier bases many times before, but to see it with images (eigenfaces!) really sold the intuition. Even better now, in this hacking life after pure math in college and grad school, is that I can build intuition now not just by proofs and exercises, but also efficient, coded implementation. Gives a different feel for the tools and concepts.