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.

What we compute

The matrix \(\mathbf{B}\) of \(T\) relative to your bases, via the change-of-basis³ formula:

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

\(\mathbf{V}\) — 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 \(u_1,\ldots,u_n\).

Applies \(T\) directly in basis coordinates:

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

If both bases are standard, \(\mathbf{B}=\mathbf{A}\).

Domain dim \(m\): Codomain dim \(n\):

Enter \(m, n\) and click Generate Matrices — or hit Load Example for a worked \(\mathbb{R}^3\!\to\!\mathbb{R}^2\) case.

Display values as: Fraction Decimal

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.

A linear transformation \(T: \mathbb{R}^m \to \mathbb{R}^n\) is a function satisfying \(T(\alpha \mathbf{u} + \beta \mathbf{v}) = \alpha T(\mathbf{u}) + \beta T(\mathbf{v})\) for all vectors \(\mathbf{u}, \mathbf{v}\) and scalars \(\alpha, \beta\). Every linear transformation between finite-dimensional spaces can be represented by a matrix. ↑
The matrix representation of \(T\) relative to bases \(\mathcal{V}\) and \(\mathcal{U}\) is the unique matrix \(\mathbf{B}\) such that \([T(\mathbf{x})]_{\mathcal{U}} = \mathbf{B}[\mathbf{x}]_{\mathcal{V}}\). Different choices of bases yield different (but similar) matrix representations. ↑
A change of basis relates coordinates in one basis to coordinates in another. If \(\mathbf{V}\) is the domain basis matrix and \(\mathbf{U}\) the codomain basis matrix, then \(\mathbf{B} = \mathbf{U}^{-1}\mathbf{A}\mathbf{V}\) converts the standard matrix \(\mathbf{A}\) to the representation in the new bases. ↑
The standard matrix of a linear transformation \(T\) is the matrix \(\mathbf{A}\) whose \(j\)-th column is \(T(\mathbf{e}_j)\), where \(\mathbf{e}_j\) is the \(j\)-th standard basis vector. When both bases are standard, the matrix representation equals the standard matrix. ↑