In earlier parts of this series we’ve looked at shopping lists as both an example and a model for something that in math class is called a vector. We’ve looked at the sum of two vectors by imagining you join together two shopping lists or joining together multiple orders for food at the fast food outlet. We then looked at something that in math class is called an inner product, and related it to finding a total price for an order based on a price list. In part 8 we introduced the idea of a matrix, as an arrangement of vectors – in particular, a rectangular arrangement of test scores, where each column shows the scores of a single student, and each row shows the scores for a single test.
In the picture above, I’ve tried a three-dimensional representation. Let’s see if this one makes sense. If we think of this as a box, the left face has the weights, the front has the test scores, and the bottom has the final scores. If we imagine the box made up of 20 small cubes, with each cube standing for one particular test, we could find Jesse’s September test on the top left. That cube has a front of 60 and a left side of .2 representing its weight. You might imagine that inside of that cube is the number 12, obtained from multiplying the test score of 60 with the weight of .2 – this 12 is what Jesse’s September test contributes to Jesse’s final score. If each cube likewise contains the product of the left face with the front face, then the bottom face of the box represents the sum of all the numbers inside of the cubes above it. Jesse’s September cube contains 12, his October cube contains 14, his November cube contains 10 and his December cube contains 40, for a total of 76, his final test score.
The box has dimensions 4 by 5 by 1, and this corresponds to a left face of 4 by 1, a front face of 4 by 5 and a bottom of 1 by 5. Though the picture above looks like a solid box, I could have tricked you and made it out of a piece of cardboard like either of these:
What these three have in common is that each, if you squint the right way, can be seen to represent the three-dimensional box shown above. All hint at how the final scores derive from test scores and weights. And yet my guess is that you have strong preferences of one of these over the other two. My guess is that few if any of you will vote for (c), and that most of you will vote for (a) as the clearest representation. Now in a prior post I had introduced a notation like (b) but without giving any clear reason why.
My intent in this post is not to argue for (b) over (a) and (c), but rather how each can be usefully seen as a (cut and) flattened version of the three-dimensional version. In the next post, we’ll extend our model to full matrix multiplication.