Comment by andrewla | Hacker News Reader

Comment by andrewla | Hacker News Reader

It took until I started learning differential geometry in the form of General Relativity to arrive at this insight, even though I feel like the notion of a matrix as a linear map was drilled in pretty thoroughly. The notion of matrix multiplication as function composition was presented almost as an interesting side effect of matrix multiplication -- that is, multiplication by these rules came first, and, hey, look, they compose!

Personally I found the prospect of tensor algebra to be much more intuitive than either of these; with matrices thrown in mostly as a computational device. Even a vector (through the dot product) is just a linear function on other vectors, and the notion of function composition carries through to that and to higher-order tensors.

Covariance and contravariance are a little more complicated to completely grok, but for most applications in Euclidean space (where the metric is the identity function) the distinction is of more theoretical interest anyway.

I found this series by eigenchris helpful to understand tensors https://www.youtube.com/watch?v=8ptMTLzV4-I&list=PLJHszsWbB6...

The metric?

A metric is a distance function. Defining a metric on a space is one of ways you create a topology.

I'm not sure what the parent means by the metric being the identity function, however. The Euclidean metric is basically the hypotenuse of a triangle parameterized by two vectors. The adjacent and opposite sides of the triangle are measured to be the Euclidean norm of each vector (their length), and the hypotenuse is the shortest distance between them.

The Euclidean metric is not the only metric - you can define distance however you'd like as long as it's consistent. But I'm not sure how the identity function works as a metric, because that would map a vector to another vector, not a scalar.

In differential geometry the metric [1] is a tensor that defines the relationship of vectors in the space to vectors in the tangent space. The identity function as a metric means that you are in a locally flat space where geodesics (the path taken by traveling in a given direction) are straight lines.

A metric in a traditional metric space is a global distance function; you can use the metric tensor in a Riemannian manifold to allow integration to find the distance between two points.

[1] https://en.wikipedia.org/wiki/Metric_tensor

Ah, so that's the setting we're talking about. Thanks for explaining that.

The metric in a vector space is a dot product. If you just have one vector space, it’s not that interesting then, but if you have many vector spaces all glued together (like a tangent space to a curved surface), then looking at how the dot product varies between nearby tangent spaces tells you a lot about the surface (Gaussian curvature and so on).

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