--- title: "Linear Algebra" description: "19 topics in Mathematics, in the order they build on each other." canonical: https://lightmysky.com/learn/mathematics/areas/linear-algebra source: https://lightmysky.com/learn/mathematics/areas/linear-algebra.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Linear Algebra 19 topics in Mathematics, in the order they build on each other. Page: https://lightmysky.com/learn/mathematics/areas/linear-algebra - [Systems of Linear Equations and Their Solution Sets](https://lightmysky.com/learn/mathematics/systems-of-linear-equations-and-their-solution-sets-mt_sQL6JfR7Te): A linear system has no solution, exactly one, or infinitely many, and nothing else. Reading which case you are in, and describing an infinite solution set, is the starting point for the whole subject. - [Gaussian Elimination and Row Echelon Form](https://lightmysky.com/learn/mathematics/gaussian-elimination-and-row-echelon-form-mt_PZBce3UvRm): Reduce a system to echelon form with three row operations that never change the solution set, then read pivots, free variables and the full solution off the result. - [Matrix Multiplication and What It Represents](https://lightmysky.com/learn/mathematics/matrix-multiplication-and-what-it-represents-mt_0fJOMPlCxr): A matrix records a linear rule. Multiplying matrices composes the rules, which is why the product is defined by rows against columns and why order matters. - [The Inverse of a Matrix and When It Exists](https://lightmysky.com/learn/mathematics/the-inverse-of-a-matrix-and-when-it-exists-mt_1K6lcfMQ_b): An inverse undoes the rule a matrix applies. Only square matrices with full pivot count have one, and elimination both decides the question and computes the answer. - [Determinants and What They Measure](https://lightmysky.com/learn/mathematics/determinants-and-what-they-measure-mt_fWr2D_1BdM): The determinant is the factor by which a map scales area or volume, with a sign for orientation. It vanishes exactly when the map collapses space, which is exactly when no inverse exists. - [LU Factorisation, Pivoting and the Cost of a Solve](https://lightmysky.com/learn/mathematics/lu-factorisation-pivoting-and-the-cost-of-a-solve-mt_5Gg2v3Oxrr): Record Gaussian elimination as a product of two triangular matrices, add pivoting for stability, and count the operations each stage costs. - [Vector Spaces and Subspaces](https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n): Strip vectors down to the two operations and the axioms they satisfy, so that polynomials, matrices and functions count as vectors too. A subspace is a subset closed under both operations. - [Linear Independence, Span and Basis](https://lightmysky.com/learn/mathematics/linear-independence-span-and-basis-mt_5RGbip9yC9): Span is everything you can reach by combining vectors; independence says none of them is redundant. A basis is a set that is both, so every element has exactly one expression. - [Dimension and the Rank-Nullity Theorem](https://lightmysky.com/learn/mathematics/dimension-and-the-rank-nullity-theorem-mt_aMG8PcwPAR): Every basis of a space has the same size, which is its dimension. For a matrix, the rank and the nullity add to the number of columns, which explains the shape of every solution set met so far. - [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto): A linear map respects addition and scaling. Once a basis is fixed, every such map is a matrix, and the matrix columns are the images of the basis vectors. - [Change of Basis](https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484): The same map has different matrices in different bases. A change-of-basis matrix converts coordinates, and similar matrices are the same map seen from two viewpoints. - [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa): Some directions survive a map unchanged except for scaling. Finding them means solving the characteristic equation, and they explain what the map does far better than its entries do. - [Diagonalisation and Powers of a Matrix](https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX): When enough independent eigenvectors exist, the map is a diagonal matrix in the eigenbasis, which makes powers, long-run behaviour and matrix exponentials easy. - [Inner Products, Length and Orthogonality](https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W): An inner product gives a general space a notion of length and angle. Orthogonal sets are independent, and orthonormal ones make coordinates cheap to compute. - [Orthogonal Projection and Least Squares](https://lightmysky.com/learn/mathematics/orthogonal-projection-and-least-squares-mt_kGvRrYJrat): Project a vector onto a subspace to get the closest point in it. Applied to an inconsistent system, this is the least squares solution, which is where regression comes from. - [The Gram-Schmidt Process](https://lightmysky.com/learn/mathematics/the-gram-schmidt-process-mt_nkp231EA_E): Turn any basis into an orthonormal one by subtracting off projections onto what has already been built, which gives the QR factorisation used by numerical solvers. - [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk): A real symmetric matrix has real eigenvalues and an orthonormal eigenbasis. That single fact is behind quadratic forms, covariance matrices and principal axes. - [The Singular Value Decomposition](https://lightmysky.com/learn/mathematics/the-singular-value-decomposition-mt_UsUZbM9VRK): Every matrix, square or not, factors into a rotation, a scaling and another rotation. The singular values rank the directions by how much the map stretches them, which is how low-rank approximation works. - [Iterative Methods for Large Linear Systems](https://lightmysky.com/learn/mathematics/iterative-methods-for-large-linear-systems-mt_FKkxKYVGYJ): For a large sparse system, elimination costs too much and the solution is approached instead of computed. Convergence is decided by the spectrum of the iteration, and preconditioning is the act of changing that spectrum.