Cheatsheet: Abstract vector spaces

The core idea

A vector is anything you can add and scale coherently (the first lesson’s math view). Arrows and lists are two examples; functions and polynomials are two more.

Functions are vectors

(f + g)(x) = f(x) + g(x)        (add pointwise)
(c · f)(x) = c · f(x)           (scale pointwise)

Both stay inside the set of functions (closure), so functions form a vector space.

Polynomials get coordinates

Basis for polynomials of degree 3 or less: {1, x, x^2, x^3}, dimension 4.

2x^2 + 5x + 7   <->   [7, 5, 2, 0]

Adding polynomials = adding coordinate vectors: (3x^2 + 1) + (x + 2) -> [1,0,3,0] + [2,1,0,0] = [3,1,3,0] -> 3x^2 + x + 3.

The derivative is a matrix

Apply d/dx to each basis polynomial; the images are the columns:

d/dx: 1->0, x->1, x^2->2x, x^3->3x^2

D = [ 0  1  0  0 ]
    [ 0  0  2  0 ]
    [ 0  0  0  3 ]
    [ 0  0  0  0 ]

Differentiate 2x^2 + 5x + 7 (coords [7,5,2,0]):

7·[0,0,0,0] + 5·[1,0,0,0] + 2·[0,2,0,0] = [5, 4, 0, 0]  =  4x + 5

Calculus by matrix multiplication.

What makes a vector space

Any set whose addition and scaling obey the standard axioms (commutativity, distributivity, a zero, etc.). The takeaway, not the list: satisfy them and every tool from the whole track applies (spans, bases, dimension, transformations, determinant, eigenvectors, change of basis).

Why it matters for AI

ML lives in abstract vector spaces, not 2D arrows.

• Embeddings (word, sentence, image): high-dim vectors; rank by cosine similarity, compress by PCA.
• Layers: linear transformation (a matrix) + nonlinearity, in a space too big to draw.
• Function spaces: signals, kernels, neural tangent kernels treat functions as vectors.
• You can now read “latent space,” “embedding space,” “function space” as vector spaces in this sense.

Pitfalls to dodge

• A vector must be an arrow or list. No, that was the teaching model; the definition is add-and-scale.
• Coordinates are absolute. No, they depend on the chosen basis here too.
• Abstraction means new rules. No, same addition, scaling, matrices, eigenvectors.
• The axioms are the thing to learn. No, the consequence is: the whole track applies.

The one-line version

A vector is anything you can add and scale, so functions and polynomials are vectors, the derivative is a matrix, and every tool from this track works on any space that follows the two rules.