Vectors that aren't arrows, abstract vector spaces
What you’ll learn
Section titled “What you’ll learn”This is lesson 15 of Track 4 (Visual Math: Linear Algebra), the capstone. The very first lesson claimed that a vector is anything you can add and scale coherently, even something that is not an arrow or a list, and called that the deepest of the three views. This lesson makes good on it. By the end you will be able to see functions and polynomials as vectors, give a polynomial honest coordinates in the basis {1, x, x^2, x^3}, write the derivative as a matrix and differentiate by matrix multiplication, and state the one idea that ties the track together: any set that obeys the add-and-scale rules is a vector space, and every tool you built (spans, transformations, determinants, eigenvectors, change of basis) applies to it unchanged. That is why the geometric intuition carries into the high-dimensional spaces where AI actually lives. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.
Where this fits
Section titled “Where this fits”This is lesson 15 of 15, the final lesson of Phase 3 (advanced perspectives) and of the whole track. It closes the loop opened in lesson 1: the “math view” the opener flagged is exactly what this capstone makes concrete. It reuses nearly every earlier lesson (basis and dimension, linear transformations, change of basis, eigenvectors) on objects that are not arrows. After this, the track’s purpose is served: the math is no longer the blocker for the technical AI tracks ahead (neural network mechanics, transformers, and the models that run on high-dimensional vector spaces).
Before you start
Section titled “Before you start”Prerequisites: ideally the whole track, since this lesson pulls earlier ideas together. The most directly reused are Spans and basis (basis, dimension), Linear transformations as moves (a linear map is captured by where it sends the basis), Change of basis, and Eigenvectors and eigenvalues. A little comfort with the derivative of a polynomial helps for the worked example, but the calculus stays at the d/dx(x^n) = n·x^(n-1) level. The practice is pen and paper.
About the math
Section titled “About the math”The arithmetic is gentle and concrete: convert polynomials to coordinate vectors, add them, and differentiate one by multiplying by the derivative matrix. The conceptual reach is the real content, seeing that addition, scaling, bases, matrices, and eigenvectors all work on functions and polynomials exactly as they did on arrows. The reward is the capstone realization that the entire track was about the rules, not the pictures.
By the end, you’ll be able to
Section titled “By the end, you’ll be able to”- Explain why functions and polynomials qualify as vectors under the add-and-scale definition
- Represent a polynomial as a coordinate vector in the basis
{1, x, x^2, x^3}and add polynomials by adding coordinates - Build the matrix of the derivative and differentiate a polynomial by matrix-vector multiplication
- State what makes a set a vector space and why every tool from the track then applies to it
- Connect abstract vector spaces to embeddings, latent spaces, and function spaces in machine learning
Time and difficulty
Section titled “Time and difficulty”- Read time: about 12 minutes
- Practice time: about 15 minutes (a polynomial-coordinates-and-differentiation exercise, an eigenvectors-of-the-derivative drill, and flashcards)
- Difficulty: standard (the capstone; light arithmetic, with the depth in seeing the whole track generalize beyond arrows)