Change of Coordinates Visualizer

A basis of \(\mathbb{R}^n\) is a set of \(n\) linearly independent vectors. Given two ordered bases

\( \mathcal{V} = \{v_1, \ldots, v_n\}, \quad \mathcal{U} = \{u_1, \ldots, u_n\}, \)

stack the basis vectors as columns of \(\mathbf{V} = [\,v_1\ \cdots\ v_n\,]\) and \(\mathbf{U} = [\,u_1\ \cdots\ u_n\,]\), each expressed in standard coordinates.

Change of coordinates matrix:

\[ \mathbf{S} = \mathbf{U}^{-1}\,\mathbf{V}. \]

If \([\mathbf{x}]_{\mathcal{V}}\) is the coordinate vector of \(\mathbf{x}\) in basis \(\mathcal{V}\), its coordinate vector in basis \(\mathcal{U}\) is

\[ [\mathbf{x}]_{\mathcal{U}} = \mathbf{S}\,[\mathbf{x}]_{\mathcal{V}}. \]

Key ideas:

• \(\mathbf{S}\) takes coordinates in \(\mathcal{V}\) to coordinates in \(\mathcal{U}\); the same geometric point \(\mathbf{x}\) is described by two different lists of numbers.
• The inverse \(\mathbf{S}^{-1} = \mathbf{V}^{-1}\mathbf{U}\) goes the other way, from \(\mathcal{U}\) back to \(\mathcal{V}\).
• If \(\mathcal{U}\) is the standard basis, then \(\mathbf{U} = \mathbf{I}\) and \(\mathbf{S} = \mathbf{V}\): the columns of \(\mathbf{V}\) themselves convert \(\mathcal{V}\)-coordinates to standard ones.
• \(\mathbf{S}\) exists only when \(\mathbf{U}\) is invertible — equivalently, when \(\{u_1, \ldots, u_n\}\) really is a basis.

For a step-by-step numerical computation of \(\mathbf{S} = \mathbf{U}^{-1}\mathbf{V}\) with fractions and surds, use the Change of Coordinates Matrix Calculator.