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An illustrated introduction to linear algebra

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Re: An illustrated introduction to linear algebra

#51

Post by rzz3 »

egonschiele wrote: post #13
dawnofdusk wrote: post #11

I really like the second part of the blogpost but starting with Gaussian elimination is a little "mysterious" for lack of a better word. It seems more logical to start with a problem ("how to solve linear equations?" "how to find intersections of lines?"), show its solution graphically, and then present the computational method or algorithm that provides this solution. Doing it backwards is a little like teaching the…

Author here – I think you're probably right. I wrote the Gaussian elimination section more as a recap, because I figured most readers have seen Gaussian elimination before, and I was keen to get to the rest of it. I'd love to hear if other folks had trouble with this section. Maybe I need to slow it down and explain it better.

I hadn’t, and your article lost me there to be honest. You didn’t explain the what, why, or when behind it, and it didn’t make sense to me at all. That said, I’m abnormally horrible at math.

Re: An illustrated introduction to linear algebra

#52

Post by MehdiHK »

egonschiele wrote: post #14
lackoftactics wrote: post #9

Aditya Bhargava did it again. I have to say I am a fan already from the old days of Grokking Algorithms.

Thank you! I loved writing that book.

One of my favorite books! Any plans to make this series a book as well? (Will be an instant buy for me)

Re: An illustrated introduction to linear algebra

#53

Post by DwnVoteHoneyPot »

egonschiele wrote: post #13
dawnofdusk wrote: post #11

I really like the second part of the blogpost but starting with Gaussian elimination is a little "mysterious" for lack of a better word. It seems more logical to start with a problem ("how to solve linear equations?" "how to find intersections of lines?"), show its solution graphically, and then present the computational method or algorithm that provides this solution. Doing it backwards is a little like teaching the…

Author here – I think you're probably right. I wrote the Gaussian elimination section more as a recap, because I figured most readers have seen Gaussian elimination before, and I was keen to get to the rest of it. I'd love to hear if other folks had trouble with this section. Maybe I need to slow it down and explain it better.

You're assumption worked for me... I've seen gaussian elimination before (but not the linear algebra) which gave me an idea of what we were doing.

Re: An illustrated introduction to linear algebra

#54

Post by anthk »

emmelaich wrote: post #44
egonschiele wrote: post #13

Earlier quoted context omitted.

Author here – I think you're probably right. I wrote the Gaussian elimination section more as a recap, because I figured most readers have seen Gaussian elimination before, and I was keen to get to the rest of it. I'd love to hear if other folks had trouble with this section. Maybe I need to slow it down and explain it better.

To some, "Now we can add the two equations together to eliminate y: might need a little explanation. The (an) answer is that since the LHS and RHS are equal, you can choose to add or subtract them to another equation and preserve equality. If I remember correctly, substitution (isolating x or y) was introduced before this technique.

Positive proportion - negative proportion = 0.

Re: An illustrated introduction to linear algebra

#55

Post by lolive »

abirch wrote: post #39
ngriffiths wrote: post #32

I feel like it's obligatory to also drop a link to the 3blue1brown series on linear algebra, for anyone interested in learning - it is a step up from what's in this post, but these videos are brilliant and still super accessible: https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...

It’s crazy his framework is open source https://github.com/ManimCommunity/manim

He details the way he uses the framework in this video: https://www.youtube.com/watch?v=rbu7Zu5X1zI

Highly recommended !

Re: An illustrated introduction to linear algebra

#56

Post by maybewhenthesun »

egonschiele wrote: post #13
dawnofdusk wrote: post #11

I really like the second part of the blogpost but starting with Gaussian elimination is a little "mysterious" for lack of a better word. It seems more logical to start with a problem ("how to solve linear equations?" "how to find intersections of lines?"), show its solution graphically, and then present the computational method or algorithm that provides this solution. Doing it backwards is a little like teaching the…

Author here – I think you're probably right. I wrote the Gaussian elimination section more as a recap, because I figured most readers have seen Gaussian elimination before, and I was keen to get to the rest of it. I'd love to hear if other folks had trouble with this section. Maybe I need to slow it down and explain it better.

I actually really liked the gaussian elimination part. It's a term you hear often and 'demystifying' it is good imho.

Only nitpick I have is that it's a pity you use only 1 and 2 in the example with the carbs. Because of the symmetry it makes it harder to see which column/row matches which part of the vector/matrix because there's only 1s and 2s and it fits both horizontally and vertically...

Re: An illustrated introduction to linear algebra

#57

Post by thaumasiotes »

> You can pick a point that sits on the first line to meet the carb goal. You can pick a point that sits on the second line to meet the protein goal. But you need a point that sits on both lines to hit both goals.

> How would a point sit on both lines? Well, it would be where the lines cross. Since these are straight lines, the lines cross only once, which makes sense because there’s only a single milk and bread combo that would get you to exactly five grams of carbs and seven grams of protein.

Geez. It's obvious that two straight lines can only cross once. It's not obvious that there's only one combination of discrete servings of bread and milk that can hit a particular target.

(It's so non-obvious that, in the general case, it isn't even true. Elimination might give you a row with all zeros.)

The fact that the solution is unique makes sense if you realize it must sit on these two lines. It makes far less sense to explain the fact that the two lines only cross once by channeling the external knowledge that the solution is unique. How did we learn that?

Re: An illustrated introduction to linear algebra

#58

Post by thaumasiotes »

rzz3 wrote: post #51
egonschiele wrote: post #13

Earlier quoted context omitted.

Author here – I think you're probably right. I wrote the Gaussian elimination section more as a recap, because I figured most readers have seen Gaussian elimination before, and I was keen to get to the rest of it. I'd love to hear if other folks had trouble with this section. Maybe I need to slow it down and explain it better.

I hadn’t, and your article lost me there to be honest. You didn’t explain the what, why, or when behind it, and it didn’t make sense to me at all. That said, I’m abnormally horrible at math.

> You didn’t explain the what, why, or when behind it

>> The trouble starts when you have two variables, and you need to combine them in different ways to hit two different numbers. That’s when Gaussian elimination comes in.

>> In the last one we were trying to make 23 cents with nickels and pennies. Here we have two foods. One is milk, the other is bread. They both have some macros in terms of carbs and protein:

>> and now we want to figure out how many of each we need to eat to hit this target of 5 carbs and 7 protein.

Re: An illustrated introduction to linear algebra

#59

Post by barrenko »

dawnofdusk wrote: post #11

I really like the second part of the blogpost but starting with Gaussian elimination is a little "mysterious" for lack of a better word. It seems more logical to start with a problem ("how to solve linear equations?" "how to find intersections of lines?"), show its solution graphically, and then present the computational method or algorithm that provides this solution. Doing it backwards is a little like teaching the…

Or something like to the tune of "what does it mean that we can eliminate", which is still unclear to me. But a lovely article, the way you (op) introduce the column perspective and really hepful for a novice such as myself.

+ there are many textbooks on LA. Not a lot of them introduce stuff in the same order or in the same manner. I think that's part of why LA is difficult to teach, and difficult to comprehend, and maybe there is no unique way to do it, so we kinda need all the perspectives we can get.

Re: An illustrated introduction to linear algebra

#60

Post by i_don_t_know »

xwowsersx wrote: post #7

This is great. I really appreciate visual explanations and the way you build up the motivation. I'm using a few resources to learn linear algebra right now, including "The No Bullshit Guide to Linear Algebra", which has been pretty decent so far. Does anyone have other recommendations? I've found a lot of books to be too dense or academic for what I need. My goal is to develop a practical, working understanding I can…

I’ve really enjoyed this book:

Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares

https://web.stanford.edu/~boyd/vmls/

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