Linear Algebra Roadmap for AI: From Vectors to Eigenvalues
If you've avoided linear algebra because it feels like 'just an abstract math course', this page is for you. Nearly every machine learning model - from simple linear regression to the largest language models - ultimately reduces to a handful of linear algebra operations: multiplying a vector by a matrix. This roadmap connects three lessons already in our curriculum, in the order that actually makes sense, so you can walk the path from 'vector' to 'eigenvalue' with as little confusion as possible.
1. Vectors and Vector Spaces
Everything starts here: a vector is the simplest way to represent a single data point (say, the pixels of an image, or a user's features) as a numeric array in space. Vector addition/subtraction, scalar multiplication, and the idea of a vector space are the absolute prerequisite for everything that follows.
Go to lesson →2. Matrices and Matrix Operations
If a vector represents a 'state', a matrix is the tool that transforms it - rotating, scaling, or fully remapping the space. Mathematically, every layer of a neural network is nothing more than a matrix multiplication followed by a nonlinear function.
Go to lesson →3. Eigenvalues and Eigenvectors
When you apply a matrix to a space, most vectors get both stretched and rotated - but a special few only get stretched, never rotated. Those special vectors and their stretch factor (the eigenvalue) are the foundation of techniques like dimensionality reduction and Principal Component Analysis (PCA).
Go to lesson →4. What's Next: Singular Value Decomposition (SVD)
Eigendecomposition only works for square matrices; SVD generalizes the same idea to any matrix (even non-square ones, like a user-item matrix in a recommender system) - one of the most practically useful tools in applied linear algebra.
Go to lesson →Ready to start?
This path is live and trackable inside our full knowledge graph - with interactive exercises and real progress tracking.