Changing Domain and Codomain Bases Independently

Two Coordinate Changes for One Map

A linear map takes vectors in a domain to vectors in a codomain. The coordinates used to describe those vectors do not have to come from the same basis, or even from spaces of the same dimension. When the domain basis changes, the input coordinates must be converted before applying the map’s matrix. When the codomain basis changes, the output coordinates must be converted afterward. These are separate operations, represented by separate transition matrices.

We work over \(\mathbb{R}\). Let \(V\) and \(W\) be finite-dimensional vector spaces, and let \(T:V\to W\) be linear. Write \(\mathcal{B}\) and \(\mathcal{C}\) for the bases used to represent the map initially, and \(\mathcal{B}'\) and \(\mathcal{C}'\) for the new domain and codomain bases. For a basis \(\mathcal{B}\), the coordinate vector \([\mathbf{x}]_{\mathcal{B}}\) records the coefficients of \(\mathbf{x}\) in that basis. The matrix \([T]_{\mathcal{C}\leftarrow\mathcal{B}}\) is defined by the rule that multiplying it by \([\mathbf{x}]_{\mathcal{B}}\) gives \([T(\mathbf{x})]_{\mathcal{C}}\).

The arrow in the subscript indicates the direction of coordinate conversion: \(\mathcal{B}'\)-coordinates go in, and \(\mathcal{B}\)-coordinates come out. In particular, it is not a direction for moving the vector itself. The same convention applies to the codomain transition matrix \(\mathbf{P}_{\mathcal{C}'\leftarrow\mathcal{C}}\), which converts \(\mathcal{C}\)-coordinates into \(\mathcal{C}'\)-coordinates.

The Independent-Basis Conversion Formula

Proof. Take any \(\mathbf{x}\in V\). First convert its new domain coordinates to the original domain coordinates: \([\mathbf{x}]_{\mathcal{B}}=\mathbf{P}_{\mathcal{B}\leftarrow\mathcal{B}'}[\mathbf{x}]_{\mathcal{B}'}\). By the definition of the original matrix of \(T\), \([T(\mathbf{x})]_{\mathcal{C}}=[T]_{\mathcal{C}\leftarrow\mathcal{B}}[\mathbf{x}]_{\mathcal{B}}\). Convert these output coordinates to the new codomain coordinates: \([T(\mathbf{x})]_{\mathcal{C}'}=\mathbf{P}_{\mathcal{C}'\leftarrow\mathcal{C}}[T(\mathbf{x})]_{\mathcal{C}}\). Substituting the first two equalities into the third gives \([T(\mathbf{x})]_{\mathcal{C}'}=\mathbf{P}_{\mathcal{C}'\leftarrow\mathcal{C}}[T]_{\mathcal{C}\leftarrow\mathcal{B}}\mathbf{P}_{\mathcal{B}\leftarrow\mathcal{B}'}[\mathbf{x}]_{\mathcal{B}'}\). By definition, \([T]_{\mathcal{C}'\leftarrow\mathcal{B}'}\) is the matrix that sends \([\mathbf{x}]_{\mathcal{B}'}\) to \([T(\mathbf{x})]_{\mathcal{C}'}\). The displayed equality holds for every \(\mathbf{x}\in V\), so the two matrices are equal. \(\square\)

This formula also explains the order of multiplication. The rightmost factor acts first: it changes the input coordinates. The middle matrix applies the map in the original bases. The leftmost factor changes the output coordinates. The dimensions match: if \(\dim V=n\) and \(\dim W=m\), the factors have sizes \(m\times m\), \(m\times n\), and \(n\times n\), so the result is an \(m\times n\) matrix.

Worked Map from \(\mathbb{R}^2\) to \(\mathbb{R}^3\)

Check the New Matrix on an Input

In general, reconstruction is a useful check because a coordinate vector is not itself the vector it describes. For example, the coordinate vector \((2,0,2)\) in this codomain basis describes \((2,0,2)\), but that agreement is specific to these coefficients and this basis. The reconstruction step must use the actual basis vectors, not treat the coordinates as standard coordinates automatically.

Recovering the Original Matrix

Common Pitfall: Confusing Coordinates with Vectors

A common error is to assign output coordinates without reconstructing the output, or to use the basis vectors as if they were standard coordinate directions. For the codomain basis \(\mathcal{C}'=((1,0,0),(1,1,0),(0,0,1))\), consider \(T(1,1)=(3,1,0)\). A mistaken claim is that its \(\mathcal{C}'\)-coordinates are \((1,\tfrac12,0)\). Reconstructing that claim gives \(1(1,0,0)+\tfrac12(1,1,0)+0(0,0,1)=(\tfrac32,\tfrac12,0)\), which is not \((3,1,0)\). The correct coordinates are \((2,1,0)\), because \(2(1,0,0)+1(1,1,0)+0(0,0,1)=(3,1,0)\).

Another frequent error is to reverse just one transition matrix. If the columns of \(\mathbf{P}_{\mathcal{C}\leftarrow\mathcal{C}'}\) are the new codomain basis vectors in old coordinates, that matrix converts new coordinates to old ones. To convert old output coordinates to new ones, use its inverse, \(\mathbf{P}_{\mathcal{C}'\leftarrow\mathcal{C}}\). The subscripts and the direction of the conversion should be checked before multiplying.

For a map from \(\mathbb{R}^n\) to \(\mathbb{R}^m\), the domain transition matrix multiplies on the right and the codomain transition matrix multiplies on the left. These operations are independent: changing only the domain basis uses the right factor, while changing only the codomain basis uses the left factor. In general, when the domain and codomain are different spaces, the two transition matrices need not be the same.