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 we compute

The change-of-coordinates matrix \(\mathbf{S}\) from \(\mathcal{V}\) to \(\mathcal{U}\):

$ \mathbf{S} = \mathbf{U}^{-1}\mathbf{V} $

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.

Step 1: Set the matrix dimension

Enter \(n\) and click Generate Matrices — or hit Load Example for a worked \(2\times 2\) case.

Display values as:
Row-reduces \([\mathbf{U}\mid\mathbf{V}]\) to \([\mathbf{I}\mid\mathbf{S}]\), showing every elementary row operation.
Resets dimensions, matrix entries, and results.