In the eyes of my mentors, to whom I’d described this work, and even showed them the manuscript, I’d simply “wasted my time”, merely doing over again something that was “already known”. But I don’t recall feeling any sense of disappointment.
A few days ago, I was thinking about matrices and determinants. I noticed that I know the formula for the determinant, but I still lack the feeling of what the determinant is. I played with that thought for some time, and then it occurred to me, that if you imagine the rows in the matrix as vectors in n-dimensional space, then the determinant of that matrix is the volume of the n-dimensional body whose edges are those vectors.
And suddenly it all made a fucking sense. The determinant is zero when the vectors are linearly dependent? Of course, that means that the n-dimensional body has been flattened into n-1 dimensions (or less), and therefore its volume is zero. The determinant doesn’t change if you add a multiple of a row to some other row? Of course, that means moving the “top” of the n-dimensional body in a direction parallel to the “bottom”, so that neither the bottom nor the height changes; of course the volume (defined as the area of the bottom multiplied by the height) stays the same. What about the determinant being negative? Oh, that just means whether the edges are “clockwise” or “counter-clockwise” in the n-dimensional space. It all makes perfect sense!
Then I checked Wikipedia… and yeah, it was already there. So much for my Nobel prize.
But it still felt fucking good. (And if I am not too lazy, one day I may write a blog article about it.)
Reinventing the wheel is not a waste of time. I will probably remember this forever, and the words “determinant of the matrix” will never feel the same. Who knows, maybe this will help me figure out something else later. And if I keep doing that, hypothetically speaking, some of those discoveries might even be original.
(The practical problem is that none of this can pay my bills.)
I kind of envy that you figured this out yourself — I learned the parallelipiped hypervolume interpretation of the determinant from browsing forums (probably this MSE question’s responses). Also, please do write that blog article.
And if I keep doing that, hypothetically speaking, some of those discoveries might even be original.
Yeah, I hope you will! I’m reminded of what Scott Aaronson said recently:
When I was a kid, I too started by rediscovering things (like the integral for the length of a curve) that were centuries old, then rediscovering things (like an efficient algorithm for isotonic regression) that were decades old, then rediscovering things (like BQP⊆PP) that were about a year old … until I finally started discovering things (like the collision lower bound) that were zero years old. This is the way.
A few days ago, I was thinking about matrices and determinants. I noticed that I know the formula for the determinant, but I still lack the feeling of what the determinant is. I played with that thought for some time, and then it occurred to me, that if you imagine the rows in the matrix as vectors in n-dimensional space, then the determinant of that matrix is the volume of the n-dimensional body whose edges are those vectors.
And suddenly it all made a fucking sense. The determinant is zero when the vectors are linearly dependent? Of course, that means that the n-dimensional body has been flattened into n-1 dimensions (or less), and therefore its volume is zero. The determinant doesn’t change if you add a multiple of a row to some other row? Of course, that means moving the “top” of the n-dimensional body in a direction parallel to the “bottom”, so that neither the bottom nor the height changes; of course the volume (defined as the area of the bottom multiplied by the height) stays the same. What about the determinant being negative? Oh, that just means whether the edges are “clockwise” or “counter-clockwise” in the n-dimensional space. It all makes perfect sense!
Then I checked Wikipedia… and yeah, it was already there. So much for my Nobel prize.
But it still felt fucking good. (And if I am not too lazy, one day I may write a blog article about it.)
Reinventing the wheel is not a waste of time. I will probably remember this forever, and the words “determinant of the matrix” will never feel the same. Who knows, maybe this will help me figure out something else later. And if I keep doing that, hypothetically speaking, some of those discoveries might even be original.
(The practical problem is that none of this can pay my bills.)
I kind of envy that you figured this out yourself — I learned the parallelipiped hypervolume interpretation of the determinant from browsing forums (probably this MSE question’s responses). Also, please do write that blog article.
Yeah, I hope you will! I’m reminded of what Scott Aaronson said recently: