Linear Algebra
Vectors, matrices, and the linear transformations that move them around. The geometric language behind data science, machine learning, computer graphics, and most of modern physics — and a subject where the right mental picture changes everything.
Vectors
Arrows with magnitude and direction. Adding, scaling, and the two ways to read a vector: as a point and as a displacement.
Dot Product
A way to multiply two vectors that returns a number. What it measures, and the surprising connection to the angle between them.
Matrices
Grids of numbers — but more usefully, encodings of linear transformations. Addition, multiplication, and what multiplication actually means.
Linear Transformations
Functions that preserve the linear structure. Rotations, reflections, scalings, and the matrix that represents each.
Determinants
A single number that captures how a matrix stretches or shrinks space — and whether it flips orientation.
Eigenvalues & Eigenvectors
The directions a linear transformation merely stretches without rotating. The single most useful concept in applied linear algebra.
Vector Spaces
The abstract definition. Subspaces, basis, dimension, coordinates, change of basis, and the four fundamental subspaces of a matrix.
Inner Product Spaces
Generalizing the dot product. Cauchy–Schwarz, orthogonal projection, and Gram–Schmidt — plus a glimpse at function inner products.
Matrix Decompositions
LU, QR, eigendecomposition, and SVD — four ways to take a matrix apart, and the applications (least squares, PCA, compression) each unlocks.