--- title: "Change of Basis" description: "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." canonical: https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484 source: https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484.md retrieved: 2026-11-12 --- > **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`. # Change of Basis 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. Subject: Mathematics · Area: Linear Algebra · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484 ## Ready when they can - Convert coordinates between two bases with a transition matrix - Write the matrix of a map in a new basis by conjugating - Explain what stays the same about a map when the basis changes ## Lesson: Same map, new coordinates The same vector has different coordinates in different bases, and the transition matrix is the dictionary between the languages. Its columns are the old basis vectors written in the new coordinates. Example: (5, 1) = 3(1, 1) + 2(1, minus 1), so its B-coordinates are (3, 2). Label every vector with its basis. **Example.** The same map has different matrices too: A new = P inverse A P, read as translate, apply, translate back. Here P has the new basis vectors as its columns, so P turns new coordinates into old ones. With A = [[1, 2], [0, 3]] and P = [[1, 1], [0, 1]], the new matrix is [[1, 0], [0, 3]], whose (1, 2) entry is 0. Conversion runs through an inverse, so it works in both directions. **Example.** To fit a projection, use a basis along its line and across it. Projection onto the line y = x keeps (1, 1) and sends (1, minus 1) to zero. In the basis (1, 1), (1, minus 1), those images have new coordinates (1, 0) and (0, 0), so the matrix is [[1, 0], [0, 0]]: keep the first coordinate, drop the second. In standard coordinates the same map is [[0.5, 0.5], [0.5, 0.5]]. *(drawing: Each basis vector's image, written in the new basis, becomes one column.)* The map stays fixed while only its description changes, so some quantities are immune. The trace, the sum of the diagonal entries, and the determinant belong to the transformation itself. So do eigenvalues: a number lambda is an eigenvalue when A v = lambda v for some nonzero v, which describes what the map does, not how the matrix is written. The diagonal matrix [[3, 0], [0, 8]] has trace 3 + 8 = 11 in every basis, and no basis change moves its determinant, 24. **Tip.** Most change of basis errors come from untracked labels. Write the basis next to every coordinate vector before you multiply. If a result looks strange, check first whether two languages got mixed in one equation. **Recap.** Convert coordinates with P, move matrices with P inverse A P, and trust trace, determinant, and eigenvalues to survive the trip. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto) ## Optional connections - [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) ## Opens up - [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa)