What is Linear Algebra?
The mathematics of space and transformation. Linear algebra gives us the language to describe position, direction, and change — the hidden engine behind graphics, AI, and modern engineering.
The Language of Space
Vectors describe position, direction, and magnitude. They are the fundamental building blocks of linear algebra — arrows that point somewhere in space with a specific length. Every physical quantity from force to velocity can be represented as a vector.
Vectors
A vector is an ordered list of numbers — coordinates that pinpoint a location or describe a direction. In 2D, a vector has two components: x and y.
Adding Vectors
Combine forces, velocities, and directions by adding vectors component by component. The tip-to-tail rule gives a beautiful geometric picture.
Matrices
Grids of numbers that transform space. A matrix takes vectors as input and produces new vectors as output — stretching, rotating, or reflecting the entire space.
Transformations
Matrices encode geometric operations. Every transformation preserves the grid lines as parallel and evenly spaced.
- Rotation — spin space around the origin
- Scaling — stretch or compress along axes
- Reflection — mirror across a line
- Shearing — slant space in one direction
The Central Problem
Given a matrix A and a result b, find the vector x. This single equation — solving systems of equations — is the beating heart of linear algebra.
A Brief History
Linear algebra evolved over centuries, shaped by mathematicians seeking to solve systems of equations and understand geometry.
3D Graphics
Every frame in every game uses matrix transformations. Rotating a character, projecting a 3D scene onto your 2D screen, lighting and shadows — all linear algebra, computed billions of times per second on your GPU.
Google PageRank
Linear algebra ranks every webpage on the internet. Google’s PageRank algorithm models the web as a giant matrix of links and finds the dominant eigenvector — the steady-state importance of every page.
LA Everywhere
Linear algebra is the universal language of applied mathematics. It appears wherever data, space, or systems interact.
- Computer graphics — rendering, animation, shaders
- Machine learning — neural networks are matrix operations
- Physics — quantum mechanics lives in vector spaces
- Engineering — circuit analysis, structural modeling
- Data science — PCA, regression, dimensionality reduction
What you learned
Vectors describe space. Matrices transform it. The equation Ax = b connects them. From ancient determinants to modern AI — linear algebra is the mathematics that makes it all work.