Graphical Linear Algebra Post 4 on 2019-02-14

This is the 4th 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 the concepts of Associativity, Zero, and Unit. 


Still the same Episode 5 – Spoilers, Adding (Part 2) and Zero

From the last post we ended talking about how composition acts on generators. The last example I gave was this one.

Fig 1

Pawel covered more examples in Episode 5 which I will replicate here. I wrote this series primarily more for my own learning. To consolidate my understanding better of this, I will replicate more of the examples Pawel showed in the same episode.

More Examples of Composition

Fig 2.1 (3,2) ; (2,1) = (3,1)

and

Fig 2.2 Also (3,2) ; (2,1) = (3,1)

This brings us to the next thing Pawel wanted his readers to learn.

Associativity

Associativity is just a technical term to describe where you put the brackets doesn’t matter for that operation. The classic example is addition. From my own memory of my secondary school mathematics textbook,

(x + y) + z = x + (y + z)

In fact, Pawel used the same concept involving addition to explain the associativity of composition with more diagrams.

The key diagrams are:

Fig 3.1 Proving the associativity based on the previous 2 diagrams

Using the (x + y) + z = x + (y + z) example to hammer in the associativity concept.

Fig 3.2 This is (x + y) + z
Fig 3.3 This is x + (y + z)

Just like how for Commutativity, Pawel showed a (I called it a blackboxed representation) of Associativity.

Fig 4.1 A reminder of the “blackboxed” representation of commutativity
Fig 4.2 Similarly, this is for associativity

Zero

Pawel goes into a long spiel about how zero is one of those rare, great scientific breakthroughs. Once again, I enjoyed those long spiels very much. But because, this series of posts aims to focus only on the absolute essentials. I have to skip all of that.

The key thing about zero, Pawel told us, is that it’s an additive identity.

Zero is the Additive Identity

Fig 5: A graphical representation of zero

And how it works can be best understood with an example.

Fig 6.1 : How zero as part of a generator to equal (1,1)
Fig 6.2: Same generator but with algebra in it

Unit – the Third Equation About Addition

Just like how we have the diagrams for Commutativity, and Associativity, the diagram 6.1 shows Unit. Now, I feel that this should be made clearer. And it doesn’t help that I have never come across this term in my old mathematics textbook. So, this is one of those “just accept it as it is for now” concepts.

In fact, Pawel says that Commutativity, Associativity, and Unit are the three equations we need to learn about addition. He also adds another point after that. Any structure (what’s an example of structure?) that satisfies the three properties of Commutativity, Associativity, and Unit are called commutative monoids. There is a link to the wikipedia definition to commutative monoids. But, trust me. It doesn’t really explain anything if you don’t math well. And I don’t math well.

Conclusion and Looking Ahead

Let’s sum up what I have learned in today’s post.

  • Associativity via examples of composition
  • The diagram for Associativity
  • Zero is the additive identity
  • The diagram for Unit (though I’m still not sure what Unit means)
  • The term commutative monoid
  • And that it’s the name for structures that satisfies all three properties of Commutativity, Associativity, and Unit

Pawel ends Episode 5 talking about using only diagrams to prove the following.

Fig 7: The upside down Unit.

But, when I look ahead at Episode 6, I must say I find his examples involving a cooking recipe far more interesting than the proof of Fig 7. I guess, that has more to do with me being a developer. In any case, I will study that in greater detail for tomorrow. See you tomorrow.

This is post #10 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.

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