Next minor, fill-in, mathematical insight...
I think what threw me off on this topic at an early age was that when I became obsessed with higher math in high school, I began looking at more advanced math books from the school library, but I didn't read all the explanatory and/or historical material that supported that math. When I came across the topic of matrices, which was never taught in my high school, I thought they were a cool idea. Basically matrices are a type of data structure, which programmers learn are basically empty slots arranged in different shapes, where you can fill in values in those slots. In the case of matrices, the structures are in the form of rectangles with rows of cells in them. However, I hadn't yet taken any programming classes, either. In any case, I thought matrices were a cool idea, since instead of just moving around numbers one at a time, you can move entire arrays of them at a time, and do useful things with them that way, basically parallel operations. All told, my perception of matrices was that they were just somebody's random idea that it would be cool to put numbers into a regular structure, then to find out if that structure had any useful mathematical properties, which it did, as is often the case. As a result, I never asked myself what a matrix really is.
I got more thrown off when I took math classes in college, and they taught us the method of Gaussian elimination, with the slightly extended version of that method called Gauss-Jordan elimination. Those methods are extremely useful in practice, easy to understand, and easy to program on a computer. However, those methods are atypical of matrix operations because they move the *rows* of a matrix around, not the columns. That gave me the erroneous impression that the essence of a matrix was a collection of stacked rows. You can see such row manipulations of the matrix in this video...
Algebra 55 - Gauss-Jordan Elimination
Apr 16, 2016
MyWhyU
It wasn't until last week that I began to rethink this. A clue is in the above video--note that they extend the idea of the matrix by adding a *column* on the right-hand side of it, not a row. They call the result an 'augmented matrix.' This suggests that matrices can be extended naturally by adding columns. After seeing the following video I realized that a matrix is more typically a collection of *columns*, not rows...
Linear transformations and matrices | Essence of linear algebra, chapter 3
Aug 7, 2016
3Blue1Brown
That's a great video for many reasons, in my opinion. Note that they start out by showing a geometrical interpretation of a matrix, which is a transformation of vectors to vectors in 2D space, which is something else I wanted to know, then they show how those vectors [shown as *columns*, not rows], naturally fit together side-by-side in a compact form that is a matrix. Suddenly it becomes clear why vectors are formally always shown as columns in math books, not as rows. (Computer science books are the opposite, since it's easier to write text horizontally in a book and in a computer program, and in programming you don't necessarily use matrices in a mathematical way, which was emphasized in the first video I posted in a post above.) Suddenly it also becomes clear why the augmented matrix of Gaussian elimination adds a *column*, not a row, also shown in the above video.
The main takeaway from all this: it is far more intuitive to think of matrices as collections of columns, not as collections of rows, especially when considering their geometrical analogies.
That's not to say it is *always* the case that only columns are added instead of rows. One notable exception, also related to geometrical operations, is the augmented matrix for affine transformations that adds a *row* at the bottom of the matrix to make the math simpler...
https://en.wikipedia.org/wiki/Affine_transformationHowever, such exceptions seem to me to be rare. In short, I'm going to start switching my perception of arrays from collections of rows to collections of columns, which will ease my understanding of how they relate to geometry.
Additional tip: Avoid sounding ignorant by pronouncing 'affine' correctly. It is pronounced like 'ah-FINE', like 'That's ah FINE transformation y'all got there.' (
https://en.wiktionary.org/wiki/affine) I once pronounced it as 'AFF-fine' to a more advanced math student in college and felt like a dummy when he corrected me. That's a common problem with forging ahead on your own using only books. I made the same mistake in high school when I started talking to my classmates about lasers and 'ZEH-nun' flash tubes, whereupon they laughed and said 'You mean ZEE-non?' Now in this day and age we can learn pronunciations automatically from YouTube videos, assuming that we're not too lazy to watch technical videos, that the videotaped teachers are native English speakers without foreign accents, and that we aren't watching ignorant people talking about religion and politics with pronunciations like 'SPEE-
seez' and '
eek-o-NOM-iks'.
