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Mathematical Proof (Breakdown)

Lynn's picture

 This is a more thorough explanation of the project I presented today. I realize that I skimmed over the actual calculations I did, but I only had five minutes and wasn't certain how interested people would be. 

Anyway, here is the breakdown, along with snapshots of the pictures I used during hte presentation, and a screenshot of the file I created to check my calculations. If you find this at all interesting, take a course in Linear Algebra. It's fantastic. I'm shamelessly promoting it here. :)

The idea behind this project was to calculate the "ranking" of my courses in a way similar to that of web pages. If you'll look at the image I've attached, you'll see that I built a matrix (the big square) out of the nine courses I have taken at Bryn Mawr so far. You can think of it like a chart, sort of; across the top is course 1, 2, 3... 9, and then down the side are those courses again. When a course influenced my thinking about another course (which I was taking at the same time or after the first course), I assigned that point on the matrix a value of 1, and when it didn't, I assigned it a zero. The first row, for example, is EvoLit; it gets a 0 in the first column because EvoLit, being itself, cannot influence itself (well, debatable, but not in Math), then a 1 in the next column because it influenced my Spanish class, and so on. I filled out the entire matrix in this way.

Is this boring? I'm terrible at explaining things.

Once I'd built the matrix, I calculated the eigenvalues - in this case, eigenvalue - of the matrix. You don't really need to know what this is, except that I needed it to get the eigenvector which would provide the solution to my problem. (The eigenvalue is 1.88, if anyone's following along on the picture. I did calculate this by hand the first time, but I  double-checked with the wonderful, magical Mathematica program. This is good, because it caught a mistake I made. Oops.) If anyone is desperately curious about how to calculate an eigenvector... Take Linear Algebra. It isn't complicated, but it does require a bit of background knowledge. (This is my way of hiding that I don't have a clue how to explain it.)

From there, I used the eigenvalue to calculate the eigenvector (again, can't really explain; the picture shows the results without all the mess, anyway, so it isn't that important). I highlighted the values that matter to us. Each value, separated by a comma, is a row in a one-column vector; each row corresponds to the same row in the original matrix. So, for example, row 1 (EvoLit) corresponds to row 1 in the eigenvector, which is about 1.23. Then, all I had to to was order the values with their corresponding classes from highest to lowest, and there were my rankings! Flashy, but simple. 

This project isn't directly comparable to web rankings, of course, because my classes are ranked predominantly by number of classes they influence, as opposed to internet rankings, where a page ranks highest if it is linked *to* by the highest number of pages. But the principle holds, and I had a lot of fun doing this. I think you can see, though, why I wasn't really able to walk through the process in my five-minute presentation time.

To anyone who happens to have stuck with me this far, did this explanation make sense? I'm happy to answer any questions. :)

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Comments

dfishervan's picture

Storytelling Through Math

I just wanted to say that I really enjoyed your presentation! It was interesting having read your prior post where you brainstormed presentation ideas and see all it come to fruition. I was really impressed with the connection and your application of our class discussions to Linear Algebra (even though I didn’t quite follow the explanation you provided above, I’m sure it is sound math!). What was interesting was that as you were explaining your project, I found myself thinking back to my 9th grade math class where I first learned about matrices, probing my memory, yearning to more fully understand your connection and story. At the same time, I was so impressed that I sort of just wanted to sit back and accept whatever you were saying. I guess this is a pretty common reaction people have to outstanding yet, complex stories. It was also intriguing that in a sense, you simplified teachings from this course as well as your other classes with a mathematical proof. Throughout the semester, we’ve moved away from explaining things with hard science and I thought it was interesting that you ended by bring back science and math into the mix as an overarching organizing power.

 

Lynn's picture

 Thank you! I didn't really

 Thank you! I didn't really think of my project that way when I prepared it - didn't consider that I was returning to hard science, which we had moved beyond - but it's kind of funny, now. I am glad that someone liked it. :)

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