This is the 3rd post in a series of daily posts on my Timeboxed Challenge to learn Graphical Linear Algebra (GLA). The source material would be the 30 episodes from the Graphical Linear Algebra blog. If you prefer to read my series from the beginning, start here with the first post. For today’s post, I will cover generators, special generator names, and how direct sum and composition act on generators.
Episode 5 – Spoilers, Adding (Part 2) and Zero
I have limited time today to read and study Episode 5 completely. Most probably I have to split Episode 5 into at least two separate days of posts. Today’s post will specifically introduce two technical terms — generator and identity. Adding to it, we will see how direct sum and composition affect generators.
Generators — Another Name for Basic Diagrams
In my last post (the one with dozens of Lego pictures), I learned from Pawel Lego bricks are useful for learning about GLA. Well, they were precursors to the basic diagrams in GLA. One such basic diagram or generator, as they are called, looks familiar.
We actually have seen this in Episode 3 before. I have reproduced this as a notation in Post 1 – On Matrix Notation and Dangling Wires to explain our normal arithmetic addition sum.
So this is a generator. Remember it. Though I must say, Pawel did not explicitly state what diagram is NOT a generator. For now, let us mothball that question. Perhaps it will be answered in future episodes.
Special Generator Names
Now, we have specific names for specific kinds of generators. Not too sure how useful to learn the names, but we’ll just go with it for now.
Identity – Just Your Typical (1, 1) Generator
When you have a Magic Lego brick with 1 hole and 1 stud, this is an identity. Well, I mean, a generator that’s (1, 1).
Twist – The (2, 2) Generator that Literally has a Twist
And a (2, 2) generator called a Twist. Which is aptly named because it has a twist in its middle.
How Direct Sum Affects Generators
If you can recall how Direct Sum affects our Magic Lego bricks, you should have no issue with using Direct Sum on generators. It’s exactly what you expect.
How Composition Affects Generators
Again, just recall how composition works on Magic Lego bricks and you’ll be fine.
A gentle reminder about composition. The left studs must match the right holes for composition to take place.
Another reminder. As long as two diagrams share the same number of holes and studs, they are considered as equal.
Conclusion and Looking ahead
A quick summary:
- A basic diagram is called a generator
- Generators are what the Magic Lego bricks were referring to. For the past few episodes.
- A (1, 1) generator is called an identity.
- A (2, 2) generator with a twist in the middle is called a twist.
- If you recall direct sum on Magic Lego, it works the same way on generators.
- If you recall composition on Magic Lego, it works the same way on generators.
- Gentle reminder 1: composition has a condition where the left studs match the right holes.
- Gentle reminder 2: two diagrams are equal when they are said to have same number of holes and studs.
Looking ahead, the last part of Episode 5 is covering more composition and a new property called associativity. Finally, it ends with an explanation about zero. I’m looking forward to covering that in tomorrow’s post.
This is post #9 in my quest for publishing weekly.
As usual, all diagrams that are shown here were taken directly from Pawel’s blog at graphicallinearalgebra.net. All credit goes to him regarding the explanation of the idea. I merely gave my take on his explanation.