Eighth Week BeyondResearch Insights, Year 1

Hi! Welcome to my eighth week of BeyondResearch Insights! Today, I would like to discuss a concept that has been recurring on both sides of the program: point-wise constructions.

I have already talked about one point-wise construction in my second-ever Beyond Research Insight, specifically, a point-wise multiplication of vectors as lists. For example, multiplying two-dimensional vectors v_1 = (x_1, y_1) \quad \text{and} \quad v_2 = (x_2, y_2) in this way would give v_1 \cdot v_2 = (x_1 y_1, x_2 y_2). Notice, that this multiplication is commutative, associative, and distributive over vector addition (which is also defined point-wise) and scalar multiplication. This is because real numbers also have these properties! For instance, commutativity is satisfied as v_1 \cdot v_2 = v_2 \cdot v_1, because (x_1 y_1, x_2 y_2) = (y_1 x_1, y_2 x_2) \quad \text{as} \quad x_1 y_1 = y_1 x_1 \quad \text{and} \quad x_2 y_2 = y_2 x_2.

This illustrates nicely the power of point-wise constructions: they allow us to lift a structure on some given set Y to the set of functions from any set X to Y. Throughout the course, this fact has helped us again and again to construct various vector spaces and prove that they satisfy the vector space axioms like commutativity and associativity of addition.

For instance, when we were tasked to construct a vector space from the set of all linear functions from \mathbb{R} \to \mathbb{R} , we did so by defining addition and scalar product point-wise and thus leveraging the fact that real number addition and multiplication already satisfy the vector space axioms. In this case, a point-wise addition meant that the sum of two linear functions f: \mathbb{R} \to \mathbb{R}, \quad x \mapsto ax \quad \text{and} \quad g: \mathbb{R} \to \mathbb{R}, \quad x \mapsto bx (where a and b are real numbers) would be (f + g): \mathbb{R} \to \mathbb{R}, \quad x \mapsto ax + bx. Again, this is both associative and commutative because the addition of real numbers is as well! Similarly, the rest of the vector space axioms are also satisfied.

The sheer number of times we have come across point-wise constructions in the program was hinting towards something deeper going on. It turns out that we can not only use point-wise constructions on real numbers to create new vector spaces but that this works for \emph{any model of an algebraic theory}! That is, all mappings from a set to a model of an algebraic theory with pointwise defined operations form a model of the same algebraic theory.

This might be a little difficult to see in the linear functions example because the domain and codomain are the same, but, essentially, they make a vector space because the operations are defined pointwise on the codomain \mathbb{R} and real numbers make a vector space. Actually, \mathbb{R}^m also makes a vector space (think m-dimensional vectors), so, for instance, we can prove that linear functions from \mathbb{R}^n \to \mathbb{R}^m (or m \times n matrices) with point-wise operations make a vector space. This was a step in a previous problem, which we discussed in our last differential geometry session (which I missed). I could have explicitly proved all of the vector space axioms for m \times n matrices, but using what we have learned about point-wise constructions and previous results would have been much more efficient!

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