Linear Algebra for Programmers with Python in 21 Days

Twenty-one days. One library you write yourself. Every routine checked against somebody else's.

Every time you call a solver or reduce a dataset, something is factoring a matrix below you, and a solver can be right about the arithmetic and wrong about the answer with nothing on screen to say which. This book is about what that something does, and how to tell when it has quietly failed.

You build one library, matkit, then check every routine against NumPy, SciPy, or LAPACK. Day 1 works a two-by-two system by hand, no code at all. Day 13 turns an orthogonalization already shown to fail into a Householder QR that does not. Day 21 solves a sparse system of a hundred million entries iteratively, without forming the array in full.

Every routine was checked against an independent oracle; matkit never calls its own. 129 listings ran across 21 chapters, and 129 output blocks are pastes from those runs, not guesses. 233 figures were drawn from those same numbers. Failures are printed with their causes, never hidden.

YOUR 21-DAY PATH

Day 1 - What Linear Algebra Actually Computes: the row and column pictures, no code this day.
Day 2 - Vectors in Code: NumPy arrays, the dot product, norms, and projection, measured.
Day 3 - Matrices as Transformations: the matrix-vector product as rotation, scaling, and shear.
Day 4 - Matrix Multiplication and What It Costs: loop orders, naive versus NumPy, timed.
Day 5 - Solving Ax = b by Elimination: forward elimination and back substitution, coded.
Day 6 - Pivoting, Failure, and the First Warning Signs: partial pivoting and near-singularity.
Day 7 - LU Factorization: PA = LU, solving with a factorization, and the determinant from U.
Day 8 - Inverses, Determinants, and Why You Rarely Want Either: cost and accuracy, measured.
Day 9 - Vector Spaces, Span, and Independence: basis, dimension, and the column space.
Day 10 - The Four Fundamental Subspaces: the null space, and rank-nullity verified numerically.
Day 11 - Norms, Conditioning, and What Floating Point Does: the condition number and the Hilbert matrix.
Day 12 - Orthogonality and Projection: the projection matrix and Gram-Schmidt's measured failure.
Day 13 - QR Factorization: Householder reflectors, and QR's cost measured against LU.
Day 14 - Least Squares: the normal equations, least squares by QR, and polynomial fitting.
Day 15 - Eigenvalues and Eigenvectors: power iteration and the Rayleigh quotient, coded.
Day 16 - Diagonalization, Powers, and Markov Chains: A to the k, and a stationary distribution.
Day 17 - Symmetric Matrices, the Spectral Theorem, and Cholesky: positive-definite tests, timed.
Day 18 - How a Real Eigenvalue Solver Works: the unshifted QR algorithm and Hessenberg reduction.
Day 19 - The Singular Value Decomposition: rotate, stretch, rotate, and the pseudoinverse.
Day 20 - Low-Rank Approximation: Eckart-Young, image compression, and PCA from the SVD.
Day 21 - Sparse Matrices, Iterative Solvers, and the Capstone: CSR storage and conjugate gradient.

WHO IT IS FOR. Developers who already write Python and have used NumPy once, whether they never took a linear algebra course or took one and came out able to invert a 3x3 matrix by hand without knowing what that was for. No prior linear algebra is assumed; every idea is built from arithmetic you can check by hand first. You need no machine learning background and no GPU; every routine here runs on a laptop CPU.

Download book source code from: {GitHub}/{mohammadnsaqib}

438 pages, black and white, checked against NumPy and LAPACK throughout.