Linear Algebra
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A matrix is a device that transforms everything at once. Scaling and rotating an image, and one layer of a neural network, are both doing the same thing: matrix multiplication.
Key points
Matrix Multiplication
Multiply an m×n matrix A by an n×p matrix B and the result AB is an m×p matrix. A fully connected layer in a neural network is exactly this product, written y = Wx + b.
Linear Transformations
A matrix is a way of transforming space. Multiply a point in the plane by a 2×2 matrix and it rotates, stretches, or flips over. When a character walks across a 3D game world, the engine is running thousands of these matrix products every single frame.
The Inverse Matrix
The inverse A⁻¹ is the matrix that undoes A: multiply them together and you get I, the identity matrix. It solves a whole system of equations in one line, x = A⁻¹b. An inverse exists exactly when the determinant det(A) is not zero.
Eigenvalues and Eigenvectors
An eigenvector v is a direction that a matrix only stretches or shrinks, never turns — Av = λv, where the number λ is the eigenvalue telling you by how much. Google's PageRank is nothing more than the eigenvector belonging to the largest eigenvalue.
Principal Component Analysis (PCA)
Eigendecomposition is how you squeeze high-dimensional data — a face image with a hundred dimensions, say — down into just a few. The largest eigenvector points along the direction carrying the most information about the data. JPEG image compression runs on the same idea.
See it drawn
The determinant tells you how much that area is multiplied by. If det = 0 the area collapses and the transformation cannot be undone.
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