Graphical Linear Algebra Post 1 on 2019-02-11

Due to unexpected circumstances, I couldn’t spend more time studying the GraphicalLinearAlgebra blog. Thankfully, the first two episodes were your typical introductory material. So they touched on big picture stuff about linear algebra and category theory. I was able to peruse them quickly. I was also able to cover what a matrix is, what a graphical notation is, and the concept of dangling wires.

Episode 1 – Makelele and Linear Algebra

In the first episode, it compares linear algebra with a famous footballer ( Claude Makélélé) in a more famous team (Real Madrid). The analogy was to point out that linear algebra is the highly critical yet unsung hero in the field of science and mathematics. Much like Makélélé.

Then, it introduces matrix to end the episode. A matrix is, basically, a double array of numbers. We describe the matrix as a row x column. So if it’s 2×3 matrix, it’s a matrix with 2 rows and 3 columns producing 6 numbers.

A 2×3 matrix

Episode 2 – Methodology, Handwaving, and Diagrams

In the second episode, Pawel Sobocinski goes into more detail about his plans for the series. There are only a few key points I like about this episode that I want to highlight.

  1. He promised not to “handwave“. In fact, he asked for specific feedback when he does indulge in handwaving. Which is an admirable stance.
  2. He forewarned about having lots of diagrams. Hence, the “graphical” in Graphical Linear Algebra. Pawel promised how in future episodes, a diagram can even tell a thousand formulas.
  3. Pawel mentioned about category theory. He even added that “category theorists are comfortable with diagrams”. And going forward, he will “use category theory to do graphical linear algebra.” Not sure what that means exactly, as I have only the vaguest notion of what category theory does.

So far so good, all simple stuff. Both episodes can be glanced through without any significant loss of understanding. Now, we come to the first real challenge of learning Graphical Linear Algebra in Episode 3.

Episode 3 – Adding (Part 1) and Mr Fibonacci

In this episode, Pawel used the example of a simple addition formula to illustrate the use of graphical notation. As I’m writing this, I also found myself going back and forth between the blog and this draft. So if this sounds more like sports commentary than a proper summary, do forgive me.

Back to the summary. Pawel started with showing simple mathematics formula can be represented graphically like this.

3 + 4 = 7

Notation for 3 + 4 = 7

And then he digresses into a story on Fibonacci and the Hindu-Arabic numeric system. Which I rather enjoy to be honest.
Because I’m assuming we’re going at the level of the freshman lecture, let me be direct here. In this episode, Pawel is trying to teach us the notation for commutativity via addition.

Commutativity is this fancy term to say that

∀ x,y.  x + y = y + x   

Another way to say this out loud is for all values of x and y such that, x + y = y + x.

And the notation for the above formula can be represented as:

This means the same as ∀ x,y.  x + y = y + x   

Then Pawel did highlight one key point for graphical linear algebra.He mentioned when we have an equation between two diagrams, the number of dangling wires on the left and right of both diagrams must be the same.

If we reference the last notation, we can see how that is true. On the left diagram, we have 2 dangling wires on the left and 1 on the right. For the right diagram, we have the exact same number of dangling wires on the left and right of that diagram.

Note the left diagram and the right diagram have the same number of dangling wires

Generalized form of addition

Finally, Pawel appears to be teaching us a general form of the diagram in the Comm equation. He says that both the left and right diagrams with their dangling wires can be presented as:

This has 2 left dangling wires and 1 right dangling wire. Same as the diagrams in the above equation.

Looking ahead to Episode 4

That’s it for now. Tomorrow, I will continue with another episode. I have previewed the next episode ahead of tomorrow’s summary. It appears that I won’t be able to compress multiple episodes into 1 blogpost summary like I did here anymore. In fact, I may need posts stretching over several days to summarize my thoughts for one episode going forward.

I’ll cross that bridge when i get there. In the meantime, thank you for reading.

This is post #7 in my quest for publishing weekly.

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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