This course is genuinely applied: every mathematical idea it introduces
exists to answer one question, "what does this do to a real image?" This
chapter builds, in this course's own notation and aimed squarely at
pixels, the handful of facts from three math courses — linear-algebra,
fourier-analysis, and calculus-iii — that the rest of the course reaches
for constantly: the eigenvalues of a covariance matrix, the discrete
Fourier transform, the gradient of a multivariable function. Each one gets
stated, worked through an example, and put to use on an image before the
section ends, so that by Chapter 1 it's a tool you've used here, not one
you're meeting for the first time.
This chapter is not a survey of every fact those three courses contain — only the handful this course reaches for constantly, reframed around images specifically and taught here at the depth this course needs. Section 1 revisits scalars, vectors, matrices, and tensors, and names which one an image actually is. Section 2 fixes this course's coordinate convention, which is not the one an ordinary math graph uses: image y grows downward, not upward, because that's how a camera or a screen lays pixels out. Section 3 builds sigma notation for sums that run over two indices — a row and a column — at once. Section 4 builds the complex-number facts specifically for the magnitude/phase language Chapter 8's frequency-domain work depends on. Sections 5 and 6 are genuinely new: how NumPy's array shape and indexing convention maps onto everything the first four sections just built, and the one-sentence synthesis — an image is a matrix-valued function of pixel coordinates — that Chapter 1 spends its own length unpacking.
If a fact in this chapter feels familiar from an earlier course, that's by
design: this chapter still derives it and works it here, in this course's
own notation, aimed at a new kind of object — a grid of pixels — rather
than sending you elsewhere to find it. Deeper background and the full
proofs stay one :::review pull away in the course each topic first came
from, but nothing in this chapter depends on you opening it.