Matrix Representation of a Linear Transformation

Given a linear transformation1 \(T: \mathbb{R}^m \to \mathbb{R}^n\) with non-standard bases, compute the matrix representation2 relative to these bases.
Uses the change of basis3 formula \(\mathbf{B} = \mathbf{U}^{-1}\mathbf{A}\mathbf{V}\).

Let \(\mathcal{V} = \{v_1, \ldots, v_m\}\) be a basis for the domain \(\mathbb{R}^m\) and \(\mathcal{U} = \{u_1, \ldots, u_n\}\) be a basis for the codomain \(\mathbb{R}^n\).

The matrix \(\mathbf{V}\) has columns \(v_1, \ldots, v_m\) (domain basis), \(\mathbf{U}\) has columns \(u_1, \ldots, u_n\) (codomain basis), and \(\mathbf{A}\) is the standard matrix4 of \(T\).

Matrix Representation:

\[ \mathbf{B} = \mathbf{U}^{-1} \mathbf{A} \mathbf{V} \]

If \([\mathbf{x}]_{\mathcal{V}}\) are coordinates in basis \(\mathcal{V}\), then:

\[ [T(\mathbf{x})]_{\mathcal{U}} = \mathbf{B} \, [\mathbf{x}]_{\mathcal{V}} \]

Key Ideas:

  • \(\mathbf{B}\) represents the transformation in non-standard bases
  • When both bases are standard, \(\mathbf{B} = \mathbf{A}\) (the standard matrix)
  • This is a similarity transformation when domain = codomain
Set matrix dimensions

Enter dimensions and click Generate Matrices to create the input grids.

Display values as:
1 — Set Up
Loads a sample transformation with m = 3, n = 2 — ready to compute.
2 — Compute
Computes the matrix representation B = U⁻¹AV via the change-of-basis formula.
Resets all dimensions, matrix entries, and results.