JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Where It All Goes: Data, Graphics, and ML

The capstone. We retrace the whole ladder — vectors to eigenvalues to the [[singular-value-decomposition|SVD]] — and watch it run real systems: graphics transforms, recommender engines via [[low-rank-approximation|low-rank approximation]], and regression via [[least-squares|least squares]]. We close with the numerical reality of the [[condition-number|condition number]] and where to go next.

The whole ladder, in one breath

You started with vectors and matrices as machines that transform space. You learned to solve systems by elimination, to spot independence, basis, and rank, to read transformations through eigenvalues and eigenvectors, and to handle imperfect data with orthogonality, projection, and least squares. This rung tied it together: every transformation can be factored into simpler pieces.

Three real systems you have touched

Graphics and games. Every rotation, scaling, and camera move is a matrix multiply. Animating a character is composing transformations frame by frame — the very composition you learned, running millions of times a second on a GPU.

Recommender systems. A giant users-by-items rating table is mostly empty. Low-rank approximation (via the SVD) assumes a few hidden taste factors explain most ratings, then fills the blanks. That is how 'people who liked this also liked...' works.

Regression and ML. Fitting a line or a linear model to noisy data is least squares — solved stably with QR. It is the first model in almost every statistics and machine-learning course, and the foundation under far fancier ones.

graphics      : output = R * S * v        (compose transforms)
recommender   : ratings ~= U_k * S_k * V_k^T (low-rank fill-in)
regression    : min || A*x - b ||  via  A = Q*R
One toolkit, three industries — the same factorizations under the hood.

The numerical reality: condition number

On paper, A*x = b either has a solution or it does not. On a real computer, every number carries tiny rounding error, and some problems amplify that error wildly. The condition number measures how much — it is the ratio of the largest singular value to the smallest, sigma_max / sigma_min.

Where to go next

You now have the full toolkit. Natural next steps: numerical linear algebra (how these factorizations are computed fast and stably at scale), optimization (where positive definite matrices and gradients take over), and machine learning, where every layer of every model is, underneath, the linear algebra you just mastered.