What we compute
The matrix \(\mathbf{B}\) of \(T\) relative to your bases, via the change-of-basis3 formula:
\[ \mathbf{B} = \mathbf{U}^{-1}\mathbf{A}\mathbf{V} \]
Matrix Representation of a Linear Transformation
Given a linear transformation1 \(T: \mathbb{R}^m \to \mathbb{R}^n\), compute its matrix representation2 in chosen bases.
Concept summary
What each input means
\(\mathbf{V}\) — domain basis. Columns are \(v_1,\ldots,v_m\).
\(\mathbf{A}\) — standard matrix4 of \(T\) (acts on standard coordinates).
\(\mathbf{U}\) — codomain basis. Columns are \(u_1,\ldots,u_n\).
What \(\mathbf{B}\) does
Applies \(T\) directly in basis coordinates:
\[ [T(\mathbf{x})]_{\mathcal{U}} = \mathbf{B}\,[\mathbf{x}]_{\mathcal{V}} \]
If both bases are standard, \(\mathbf{B}=\mathbf{A}\).
Enter \(m, n\) and click Generate Matrices — or hit Load Example for a worked \(\mathbb{R}^3\!\to\!\mathbb{R}^2\) case.
Step-by-step: first \(\mathbf{U}^{-1}\), then \(\mathbf{U}^{-1}\mathbf{A}\), then \(\mathbf{B} = \mathbf{U}^{-1}\mathbf{A}\mathbf{V}\).
Resets dimensions, matrix entries, and results.
Cite this tool
Kapita, S. (2026). Matrix Representation of a Linear Transformation. Math Tools. https://doi.org/10.5281/zenodo.20981284
Kapita, Shelvean. "Matrix Representation of a Linear Transformation." Math Tools, 2026, doi.org/10.5281/zenodo.20981284.
@online{kapita2026lineartransform,
author = {Shelvean Kapita},
title = {{Matrix Representation of a Linear Transformation}},
year = {2026},
organization = {Math Tools},
doi = {10.5281/zenodo.20981284},
url = {https://doi.org/10.5281/zenodo.20981284}
}