Row operations turn many equations into a form you can solve step by step. You practice elimination, pivots, free variables, and checking whether an answer really works.

Linear algebra grew from solving equations into a language for geometry, physics, computing, statistics, and data. This chapter shows why ideas like determinants, matrices, vector spaces, and numerical algorithms appeared when they did.

Matrices organize numbers so they can store systems of equations and move vectors around. You read rows, columns, entries, and matrix-vector products as practical objects instead of decoration.

Matrix multiplication means applying one action after another. You practice the rules, see why order matters, and connect multiplication to composition of transformations.

Python notebooks give you a practical way to calculate, plot, and check your work. You use arrays, exact arithmetic when helpful, and numerical tools without letting software hide the math.

An inverse matrix reverses a linear action when reversal is possible. This chapter explains one-answer systems, singular matrices, and why solving equations is usually better than computing an inverse by hand.

Span and independence describe which vectors can build other vectors and which ones add truly new information. You practice spotting redundant vectors and describing all possible combinations.

A basis gives a space its coordinate system, and dimension counts the number of independent directions. You build bases for familiar and unfamiliar spaces, including spaces made of polynomials, functions, and matrices.

Rank and null spaces reveal what a matrix can produce and what it loses. You connect column space, row space, null space, and left null space to solvability and degrees of freedom.

Linear transformations are the coordinate-free idea behind matrices. You describe stretching, rotation, projection, shear, and collapse as actions on whole spaces.

Changing basis lets you describe the same vector or transformation in a more useful coordinate system. You practice moving between coordinates while keeping the underlying problem unchanged.

Determinants measure signed area, volume, orientation, and whether a transformation collapses space. You calculate them, interpret them, and use them carefully without treating them as magic.

Eigenvectors point in directions a transformation does not turn, only scales. You find eigenvalues and eigenvectors and connect them to stability, steady states, and simplified calculations.

Diagonalization turns repeated matrix actions into easier scalar calculations. You use it to compute powers, solve recurrence relations, and see when a coordinate system reveals the structure of a transformation.

Complex eigenvalues explain rotations, oscillations, and spirals that real numbers alone cannot describe cleanly. You handle complex vectors and translate the results back into real behavior.

Some matrices do not have enough eigenvectors to diagonalize. This chapter covers generalized eigenvectors, Jordan form, and why these ideas matter more for theory than for many numerical computations.

Inner products turn vector spaces into places where length, angle, distance, and similarity make sense. You compute dot products and work with weighted and function-based inner products.

Gram-Schmidt turns a useful but messy basis into perpendicular unit directions. You use orthogonal bases to simplify coordinates, projections, and calculations.

Projection finds the closest vector inside a chosen subspace. You use projection to separate signal from error and to make impossible systems as close to solvable as possible.

Least squares handles systems with noisy data or too many equations. You fit lines and models, read residuals, and judge whether the answer is meaningful.

QR factorization packages orthogonalization into a reliable matrix tool. You use QR to solve least-squares problems more stably than the normal equations.

Dual vectors turn vectors into measurements, and adjoints describe the matching reverse action. You connect transposes, conjugate transposes, linear functionals, and inner products.

Symmetric and Hermitian matrices have especially clean geometry: real eigenvalues and perpendicular eigenvectors. The spectral theorem becomes a powerful way to break a transformation into simple directions.

Quadratic forms describe curved surfaces, energy, variance, and optimization landscapes. You classify them using eigenvalues and work with positive definite matrices.

The singular value decomposition breaks any matrix into rotations, stretching, and projection-like pieces. You use singular values to measure rank, stability, compression, and the strength of directions in data.

Low-rank methods keep the most important structure while throwing away small details. You apply SVD to compression, denoising, and principal component analysis.

Real computations have rounding error, and some problems magnify tiny input changes. You use norms, residuals, and condition numbers to judge how much to trust an answer.

LU and Cholesky factorization solve common systems efficiently and reliably when used with the right safeguards. You practice pivoting, triangular solves, and choosing a factorization that matches the matrix.

Huge systems often cannot be solved by direct factorization. You use iterative methods such as Jacobi, Gauss-Seidel, conjugate gradient, and Krylov ideas to improve an answer step by step.

Sparse matrices store only the entries that matter, which makes large networks and scientific models possible. You connect matrices to graphs, adjacency, incidence, Laplacians, flows, and ranking.

Markov chains use matrices to model movement between states over time. You compute transition powers, steady-state distributions, absorbing behavior, and long-term predictions.

Linear algebra solves systems that evolve through time, from recurrences to differential equations. You use eigenvalues and matrix exponentials to describe growth, decay, oscillation, and stability.

Tensors extend vectors and matrices to data with more than two directions, such as images, videos, language batches, and scientific measurements. You handle tensor shapes, contractions, unfolding, and multilinear maps.

Optimization often needs derivatives of vector and matrix expressions. You compute gradients, Jacobians, Hessians, and matrix derivatives used in fitting models and solving inverse problems.

Automatic differentiation applies the chain rule through large networks of linear and nonlinear operations. You trace how matrix products, transposes, and gradients power modern deep learning software.

Modern hardware performs massive matrix and tensor operations in parallel. You reason about batching, memory layout, precision choices, GPUs, and why performance depends on more than formula counts.

Randomized methods find useful structure in matrices too large to fully inspect. You use sampling, sketching, randomized SVD, and error checks to get fast approximate answers.

Embeddings, similarity search, and attention rely heavily on dot products, matrix multiplication, normalization, and low-dimensional structure. This chapter connects core linear algebra to search, recommendation, and foundation-model workflows.

You take a messy real problem from question to model, computation, validation, and presentation. The workflow includes choosing representations, testing assumptions, checking numerical risk, and explaining results clearly.

Linear algebra opens paths into data science, engineering, graphics, physics, economics, optimization, and machine learning. You plan next steps, portfolio projects, proof-of-skill artifacts, and habits for staying current.