Graphical linear algebra: from matrices to monoids

The best, hands down on YouTube.

There's no operator because composition and direct sum are implicit in the order of your program words.

It’s not surprising—there’s a deep relationship among concatenative languages, Hughes’ arrows, combinator calculus, category theory, linear algebra, logic, even topology. I haven’t read through this but I guess the ⊕ operator would be “* * *” (parallel composition) in arrow notation; in Haskell it has the type “Arrow a ⇒ a b c → a b′ c′ → a (b, b′) (c, c′)”, or “(b → c) → (b′ → c′) → (b, b′) → (c, c′)” specialised to the function arrow.

I think the major difference is this notation has the count of loose wires corresponding to dimension, and Penrose notation has the count of loose wires corresponding to tensor valence.

Here, a diagram with 1L and 2R wires corresponds to a 2d-vector, in Penrose's notation it would correspond to a (1,2) tensor that takes kd-vectors to kXk matrices.

But... they certainly have a similar feel. I wonder if you could build up the Penrose notation out of this.

From what I can tell, this series is a kind of intro to category theory by way of linear algebra

The structure is more obviously modular and the algebraic laws become topological phenomena, which makes it easier to consider more complicated structures than vector spaces.