What we compute
The change-of-coordinates matrix \(\mathbf{S}\) from \(\mathcal{V}\) to \(\mathcal{U}\):
$ \mathbf{S} = \mathbf{U}^{-1}\mathbf{V} $
Change of Coordinates Matrix Calculator
Compute the matrix \(\mathbf{S}\) that converts coordinates from basis \(\mathcal{V}\) to basis \(\mathcal{U}\) in \(\mathbb{R}^n\). Prefer pictures? See the Coordinate Transform Visualizer.
Concept summary
What each input means
\(\mathbf{V}\) — initial basis. Columns are \(v_1,\ldots,v_n\).
\(\mathbf{U}\) — final basis. Columns are \(u_1,\ldots,u_n\).
We row-reduce \([\mathbf{U}\mid\mathbf{V}]\) to \([\mathbf{I}\mid\mathbf{S}]\).
What \(\mathbf{S}\) does
Translates coordinates between bases:
$ [\mathbf{x}]_{\mathcal{U}} = \mathbf{S}\,[\mathbf{x}]_{\mathcal{V}} $
The inverse \(\mathbf{S}^{-1}=\mathbf{V}^{-1}\mathbf{U}\) goes the other way.
Enter \(n\) and click Generate Matrices — or hit Load Example for a worked \(2\times 2\) case.
Row-reduces \([\mathbf{U}\mid\mathbf{V}]\) to \([\mathbf{I}\mid\mathbf{S}]\), showing every elementary row operation.
Resets dimensions, matrix entries, and results.
Cite this tool
Kapita, S. (2026). Change of Coordinates Matrix Calculator. Math Tools. https://doi.org/10.5281/zenodo.20981161
Kapita, Shelvean. "Change of Coordinates Matrix Calculator." Math Tools, 2026, doi.org/10.5281/zenodo.20981161.
@online{kapita2026changebasis,
author = {Shelvean Kapita},
title = {{Change of Coordinates Matrix Calculator}},
year = {2026},
organization = {Math Tools},
doi = {10.5281/zenodo.20981161},
url = {https://doi.org/10.5281/zenodo.20981161}
}