Nice article. That's how I learned matrices in high school in Germany. Maybe it's different here in the US, I'll have to take a look at my daughters' textbooks.
Relearning Matrices as Linear Functions
11–20 of 95 posts
Re: Relearning Matrices as Linear Functions
#12Having not taken a linear algebra course in college, does anyone have a recommendation for a book/course to follow?
Re: Relearning Matrices as Linear Functions
#13Nice article. That's how I learned matrices in high school in Germany. Maybe it's different here in the US, I'll have to take a look at my daughters' textbooks.
They were in the textbook in my high school, but we always skipped that chapter.
Re: Relearning Matrices as Linear Functions
#14It took until I started learning differential geometry in the form of General Relativity to arrive at this insight, even though I feel like the notion of a matrix as a linear map was drilled in pretty thoroughly. The notion of matrix multiplication as function composition was presented almost as an interesting side effect of matrix multiplication -- that is, multiplication by these rules came first, and, hey, look, they compose!
Personally I found the prospect of tensor algebra to be much more intuitive than either of these; with matrices thrown in mostly as a computational device. Even a vector (through the dot product) is just a linear function on other vectors, and the notion of function composition carries through to that and to higher-order tensors.
Covariance and contravariance are a little more complicated to completely grok, but for most applications in Euclidean space (where the metric is the identity function) the distinction is of more theoretical interest anyway.
Re: Relearning Matrices as Linear Functions
#15Hey OP here!
When I first was introduced to matrices (high school) it was in the context of systems of equations. Matrices were a shorthand for writing out the equations and happened to have interesting rules for addition etc. It took me a while to think about them as functions on their own right and not just tables. This post is my attempt to relearn them as functions which has helped me develop a much stronger intuition for linear algebra. That’s my motivation for this post and why I decided to work on it. Feedback is more than welcome.
Re: Relearning Matrices as Linear Functions
#16Good, intuitive introduction to matrices. Next steps could be showing that there are infinitely many different matrix representations of a linear map (different from the polynomials) and they can be used for function spaces, too. One question that usually pops up that I was confused about till recently: are rank two tensor equivalent to matrices? Answer is no, e.g. see here: https://physics.stackexchange.com/question…
Hey!
Thanks for the feedback. I go into this in the next post on eigenvectors here: https://www.dhruvonmath.com/2019/02/25/eigenvectors/. I start by discussing basis vectors which I believe is what you’re looking for in your comment.
Re: Relearning Matrices as Linear Functions
#17Who is "we" in this context?
Hey! I wrote this article - “we” is referring to people who had a similar educational experience to me. I was introduced to matrices as a tool for solutions to systems of equations. I always wish I was taught the functional perspective from the beginning.
Re: Relearning Matrices as Linear Functions
#18Nice! The illustrations + color coding for the vectors are very useful.
Here is a video tutorial that goes through some of the same topics (build up matrix product from the general principle of a linear function with vector inputs): https://www.youtube.com/watch?v=WfrwVMTgrfc
Associated Jupyter notebook here: https://github.com/minireference/noBSLAnotebooks/blob/master...
Re: Relearning Matrices as Linear Functions
#19Hey OP here! When I first was introduced to matrices (high school) it was in the context of systems of equations. Matrices were a shorthand for writing out the equations and happened to have interesting rules for addition etc. It took me a while to think about them as functions on their own right and not just tables. This post is my attempt to relearn them as functions which has helped me develop a much stronger intu…
What got me for a while was the concept of a tensor:
For example: What is a tensor?
Wrong way to answer it: Well, the number 5 is a tensor. So's a row vector. So's a column vector. So's the dot product and the cross product. So's a two-dimensional matrix. So's a four-dimensional matrix, just... don't ask me to write one on the board, eh? So's this Greek letter with smaller Greek letters arranged on its top right and bottom right. Literally anything you can think of is a tensor, now... try to find some conceptual unity.
Then coordinate-free fanaticism kicked in, robbing the purported explanations of any explanatory power in terms of practical applications of tensors. The only thing they could do was shift indices around.
What finally made it stick is decomposing every mathematical concept into three parts:
1. Intuition, or why we have the concept to begin with.
2. Definitions, or the axioms which "are" the concept in some formal sense.
3. Implementations, or how we write specific instances of the concept down, including things like the source code of software which implements the concept.
Re: Relearning Matrices as Linear Functions
#20Hey OP here! When I first was introduced to matrices (high school) it was in the context of systems of equations. Matrices were a shorthand for writing out the equations and happened to have interesting rules for addition etc. It took me a while to think about them as functions on their own right and not just tables. This post is my attempt to relearn them as functions which has helped me develop a much stronger intu…
>geometrically, all linear maps can be thought of as rotations and scalings.
and reflections.