Arithmetic deals in single numbers. Linear algebra deals in whole lists of numbers at once — a point in space, the pixels of an image, the features of a house, the weights of a neural network — and the clean, geometric rules for pushing those lists around. It is the mathematics of vectors, matrices, and the linear transformations that stretch, rotate and flatten space.

It is also, quietly, the most useful maths there is. Computer graphics, search engines, GPS, quantum mechanics and nearly all of machine learning run on it. The reason is simple: computers are fast at exactly one thing — multiplying big grids of numbers — and linear algebra is the art of phrasing a problem so that's all it takes.

The big idea: keep the grid straight

One thread runs through everything here. A linear map is one that keeps grid lines straight and evenly spaced, and keeps the origin fixed. That single restriction is surprisingly powerful: it means the whole transformation is pinned down by where just a couple of arrows land, and it lets us trade geometry (arrows, areas, rotations) for arithmetic (grids of numbers) and back again, freely, whenever one is easier than the other.

The shape of the journey

This course moves in six stages, each building on the last.

• Stage A — Vectors. Arrows and lists of numbers: adding, scaling, length, and the all-important dot product.
• Stage B — Vector spaces. What a bunch of vectors can reach: span, independence, basis and dimension.
• Stage C — Matrices. Grids of numbers, and the one operation that matters most — a matrix acting on a vector.
• Stage D — Linear transformations. Matrices are motions of space: scaling, rotation, shear, the determinant, and the inverse.
• Stage E — Linear systems. Many equations at once, solved by elimination or by inverting a matrix.
• Stage F — Eigenvectors & decomposition. The special directions a matrix only stretches — the key to diagonalization, symmetric matrices and the SVD.

Stage A — Vectors

1. What Is a Vector?
2. Vectors as Coordinates
3. Adding Vectors
4. Subtracting Vectors
5. Scaling a Vector
6. Linear Combinations
7. The Length of a Vector
8. Unit Vectors
9. The Dot Product
10. The Dot Product and Angle
11. Orthogonal Vectors
12. Projecting One Vector onto Another

Stage B — Vector spaces

1. Span
2. Linear Independence
3. Basis and Dimension
4. Coordinates in a Basis
5. Vectors in 3D

Stage C — Matrices

1. What Is a Matrix?
2. Adding Matrices
3. The Transpose
4. A Matrix Times a Vector
5. Multiplying Matrices
6. How Matrix Products Behave
7. The Identity Matrix
8. Diagonal and Symmetric Matrices

Stage D — Linear transformations

1. Linear Transformations
2. Transformations Are Matrices
3. Scaling and Reflection
4. Rotation Matrices
5. Shears
6. Composing Transformations
7. The Determinant
8. When a Matrix Collapses Space
9. The Inverse Matrix
10. Computing a 2×2 Inverse

Stage E — Linear systems

1. Systems of Linear Equations
2. Writing a System as Ax = b
3. Row Operations
4. Gaussian Elimination
5. Solving with the Inverse
6. Rank and How Many Solutions

Stage F — Eigenvectors & decomposition

1. Eigenvectors and Eigenvalues
2. The Characteristic Equation
3. Finding Eigenvectors
4. Diagonalization
5. Symmetric Matrices
6. The Singular Value Decomposition

Let's get started

We begin with the atom of the whole subject — the humble vector. By the end of Stage A you'll see why an arrow and a list of numbers are the same thing.

Let's get started → What Is a Vector?