Volume 0 — The Array and the Machine

Before the first matrix, three notebooks about the thing that will hold it.

A vector in a lecture is an element of a vector space. A vector in a Python session is a contiguous block of bytes with a shape attached, a rule for reading strides out of that block, and a floating-point format that cannot represent one tenth. Every result in the remaining eight volumes is produced by that object, so it is worth knowing what it can and cannot do before asking it to diagonalise anything.

The volume is short and it is not optional. §0.1 is the array model — shapes, strides, views versus copies, broadcasting, and the reason a triple loop is four orders of magnitude slower than one @. §0.2 is floating-point reality: machine epsilon, catastrophic cancellation, and the habit that replaces == with a tolerance, which is the habit every validation in this course rests on. §0.3 is the vector itself — dot products, angles, norms, projections — done concretely enough that Volume II’s least squares is a short step rather than a leap.

There is one idea here that the rest of the course keeps cashing in. A computation has two kinds of error: the error you make because your method is approximate, and the error the machine makes because it stores 53 bits. The first shrinks when you work harder. The second does not, and knowing which one you are looking at is most of numerical analysis. §0.2 puts a number on both.

The material is elementary and the pitch is not. A reader who has never met np.einsum will meet it in §0.1, and a reader who has used NumPy for years will probably still find out something uncomfortable about what A.T @ A costs. That is the intended experience: familiar tools seen from the side they are usually turned away from.