Linear Algebra
From vectors to eigenstructure.
Nine units. The math of flat spaces — and the language every modern field speaks.
Unit 1 · Building blocks
Vectors & spaces
Vectors, span, linear combinations, the geometry of ℝⁿ.
Unit 2 · SystemsSystems of equations
Row reduction, echelon form, Gaussian elimination, homogeneous vs. particular.
Unit 3 · StructureMatrix algebra
Multiplication, inverses, transpose, block matrices, the rules that make it work.
Unit 4 · GeometryDeterminants
The signed volume of a parallelepiped — and what it tells you about invertibility.
Unit 5 · SubstructureSubspaces & basis
Column space, null space, rank-nullity, change of basis.
Unit 6 · TransformationsLinear maps
Matrix representations, kernel, image, isomorphisms, similarity.
Unit 7 · Inner productsInner product spaces
Dot products, norms, orthogonality, Gram-Schmidt, projections.
Unit 8 · Eigen-theoryEigenvalues & eigenvectors
Characteristic polynomials, diagonalization, the spectral theorem for symmetric matrices.
Unit 9 · ApplicationsMatrix decompositions
LU, QR, singular value decomposition — the algorithms data science runs on.