Phase 4
Linear Algebra
Phase 4 of the Quant Academy curriculum.
4.1Full Lesson
Vector Spaces, Span, Independence, Basis, Dimension
The abstract stage on which all of linear algebra plays out: closure, spanning sets, independence, and the invariant cal
4.2Full Lesson
Linear Maps, Matrices, Rank, and the Null Space
How matrices represent linear transformations, and the rank-nullity theorem that governs solvability of linear systems.
4.3Full Lesson
Determinants, Eigenvalues, and Eigenvectors
The determinant as signed volume scaling, and eigenpairs as the invariant directions a matrix merely stretches.
4.4Full Lesson
Diagonalization and Similarity
Rewriting a matrix in an eigenbasis so that acting, powering, and exponentiating become elementwise operations.
4.5Full Lesson
Inner-Product Spaces, Orthogonality, and Projections
Geometry for abstract vectors: angles, orthonormal bases, orthogonal projection, and the least-squares solution.
4.6Full Lesson
The Spectral Theorem, Symmetric and Positive-Definite Matrices, Quadratic Forms
Why symmetric matrices are the best behaved of all: real eigenvalues, orthogonal eigenvectors, and the geometry of covar
4.7Full Lesson
The Singular Value Decomposition and Applications (PCA, Covariance)
The factorization every matrix admits, and how it powers principal component analysis of asset returns.