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This is why one of the most valuable uses of time for mathematicians and especially software engineers is to study the historical development of techniques and

by sedachv 12y ago

This is why one of the most valuable uses of time for mathematicians and especially software engineers is to study the historical development of techniques and technologies. Understanding how matrices are a product of the quest for finding solutions to systems of linear equations gives you a much better idea for when and how to apply matrix techniques. Most math textbooks limit themselves to "The determinant is defined by ad-bc" which on its own is an almost completely useless fact to know outside of taking an exam. Software engineering is much worse in terms of useless complications developed by people who don't know previous solutions (angular.js? Great! Have you ever heard of dataflow programming? Constraint satisfaction? Dynamic binding? No? No wonder angular is such a piece of shit).

thomasahle 12y ago
I agree. I also always found I learned math better when it came from authors who knew the history of the topic and where the different ideas fit in with the wider landscape of mathematics or cs. Knuth and Terence Tao are two people who do this amazingly well. And I believe, really prioritises it.
sdenton4 12y ago
To be a bit contrarian, linear algebra is waaaay more useful than just as an outgrowth of solving systems of equations. My own point of view is that linear algebra is by far the most successful part of mathematics: 'Most' questions you can come up with have satisfactory answers. This is in contrast to, say, number theory, where there's a bunch of nice elementary results and a lot of interesting questions that seem nigh impossible to solve. As a result, it's a pretty common game in mathematics to start with something new or difficult that you want to describe, and then do your level best to turn your questions into linear algebra problems so that you can actually get answers. The extent to which this doesn't work is the extent to which you need to develop new ideas. (One example of such an approach is algebraic graph theory. Turn a graph into an interesting matrix, and then use the linear algebraic properties of that matrix to describe interesting properties of your graph.)
- alicez 12y ago
  Linear algebra is one of my favorite subjects. Its design is beautifully simple, yet extremely powerful. Half of modern machine learning (and all of Matlab) is built on matrix algebra. And the existence of fast software for numeric linear algebra makes it practically applicable. The link to graph theory is beautiful, too. Entries in an matrix can represent edge weights, and taking a random walk on a graph can be represented as a matrix-vector multiplication, and the stationary distribution is the singular vector. How cool is that? When you start to link together abstractions from different fields of mathematics and science, you get these fantastic insights that are just mind bogglingly awesome. This is what makes all the pain of wading through an ocean of symbols and equations worthwhile, imho.
- sedachv 12y ago
  > My own point of view is that linear algebra is by far the most successful part of mathematics: 'Most' questions you can come up with have satisfactory answers. This is in contrast to, say, number theory, where there's a bunch of nice elementary results and a lot of interesting questions that seem nigh impossible to solve. I view that as "if you can find a way to express your problem in terms of linear equations, there are well-known techniques for finding solutions."
luddypants 12y ago
Can you suggest any good reads which incorporate the history & motivation of linear algebra? I have had some experience with the common text books, but never put much effort into incorporating it into the way I think, precisely because it did not seem worth my time to just memorize methods without much context.
- sedachv 12y ago
  I recently enjoyed Bashmakova and Smirnova's The beginnings and evolution of algebra. It covers the development of algebra from Babylonian math to group theory. One of the valuable things it does is cover the evolution of notation for mathematical problems. Now next time someone complains that "Prefix is hard! Infix is natural!" I can throw this book at them.
- sdenton4 12y ago
  'Proofs and refutations' is a fantastic book about the process of mathematical exploration, whose primary worked example ('What is a polytope? And what does Euler's Formula mean?') ends up being a non-obvious translation of geometry into linear algebra. It's a really wonderful read, and will give a bit of a sense of how mathematicians think about linear algebra, in a way that most textbooks don't.

by sedachv 12y ago

thomasahle 12y ago

I agree. I also always found I learned math better when it came from authors who knew the history of the topic and where the different ideas fit in with the wider landscape of mathematics or cs. Knuth and Terence Tao are two people who do this amazingly well. And I believe, really prioritises it.

sdenton4 12y ago

To be a bit contrarian, linear algebra is waaaay more useful than just as an outgrowth of solving systems of equations. My own point of view is that linear algebra is by far the most successful part of mathematics: 'Most' questions you can come up with have satisfactory answers. This is in contrast to, say, number theory, where there's a bunch of nice elementary results and a lot of interesting questions that seem nigh impossible to solve. As a result, it's a pretty common game in mathematics to start with something new or difficult that you want to describe, and then do your level best to turn your questions into linear algebra problems so that you can actually get answers. The extent to which this doesn't work is the extent to which you need to develop new ideas. (One example of such an approach is algebraic graph theory. Turn a graph into an interesting matrix, and then use the linear algebraic properties of that matrix to describe interesting properties of your graph.)

alicez 12y ago
Linear algebra is one of my favorite subjects. Its design is beautifully simple, yet extremely powerful. Half of modern machine learning (and all of Matlab) is built on matrix algebra. And the existence of fast software for numeric linear algebra makes it practically applicable. The link to graph theory is beautiful, too. Entries in an matrix can represent edge weights, and taking a random walk on a graph can be represented as a matrix-vector multiplication, and the stationary distribution is the singular vector. How cool is that? When you start to link together abstractions from different fields of mathematics and science, you get these fantastic insights that are just mind bogglingly awesome. This is what makes all the pain of wading through an ocean of symbols and equations worthwhile, imho.
sedachv 12y ago
> My own point of view is that linear algebra is by far the most successful part of mathematics: 'Most' questions you can come up with have satisfactory answers. This is in contrast to, say, number theory, where there's a bunch of nice elementary results and a lot of interesting questions that seem nigh impossible to solve. I view that as "if you can find a way to express your problem in terms of linear equations, there are well-known techniques for finding solutions."

luddypants 12y ago

Can you suggest any good reads which incorporate the history & motivation of linear algebra? I have had some experience with the common text books, but never put much effort into incorporating it into the way I think, precisely because it did not seem worth my time to just memorize methods without much context.

sedachv 12y ago
I recently enjoyed Bashmakova and Smirnova's The beginnings and evolution of algebra. It covers the development of algebra from Babylonian math to group theory. One of the valuable things it does is cover the evolution of notation for mathematical problems. Now next time someone complains that "Prefix is hard! Infix is natural!" I can throw this book at them.
sdenton4 12y ago
'Proofs and refutations' is a fantastic book about the process of mathematical exploration, whose primary worked example ('What is a polytope? And what does Euler's Formula mean?') ends up being a non-obvious translation of geometry into linear algebra. It's a really wonderful read, and will give a bit of a sense of how mathematicians think about linear algebra, in a way that most textbooks don't.