Plain-language explanation.
Linear algebra is the mathematics of vectors, matrices, and linear transformations. It is the language of data science, machine learning, computer graphics, quantum mechanics, and virtually every area of applied mathematics and engineering.
Core concepts and standard treatment.
Core linear algebra covers vectors and vector spaces (linear independence, span, basis, dimension), matrix operations (addition, multiplication, transpose, inverse), systems of linear equations (Gaussian elimination, row echelon form), determinants (cofactor expansion, Cramer's rule), eigenvalues and eigenvectors (characteristic polynomial, diagonalisation), and key factorizations (LU, QR, Cholesky). The four fundamental subspaces (Gilbert Strang's framework — column space, null space, row space, left null space) provide deep structural insight.
Deeper theory, debates and edge cases.
Advanced linear algebra covers singular value decomposition (SVD — principal component analysis, low-rank approximation, the Moore-Penrose pseudoinverse), spectral graph theory (graph Laplacian, algebraic connectivity — Cheeger's inequality), tensor algebra (multilinear maps, Tucker decomposition, tensor networks — quantum computing), numerical linear algebra (condition number, stability, iterative solvers — conjugate gradient, GMRES), and functional analysis (Hilbert spaces, bounded operators, spectral theorem in infinite dimensions).
How it is applied in practice.
At the data scientist, ML engineer, and quantitative researcher level, linear algebra is the fundamental toolkit: PCA and SVD for dimensionality reduction (scikit-learn, numpy.linalg); matrix factorisation for recommendation systems (collaborative filtering); convolutional neural networks as linear maps composed with non-linearities (PyTorch, JAX); quantum circuit representation as unitary matrices; portfolio optimisation as quadratic programming (covariance matrix inversion — Markowitz); and numerical PDE solvers (sparse matrix methods — SciPy sparse).