Mixed-Basis Representation Workshop

Tracking Bases Through a Polynomial Map

A matrix for a linear map is tied to choices of bases: its input is a coordinate column in the domain basis, and its output is a coordinate column in the codomain basis. When those bases differ, the entries still follow a consistent rule, but it is important to keep track of which coordinates a matrix accepts and produces. We will do that explicitly for maps between polynomial spaces, then use the same bookkeeping for an action, a composition, and a sum.

Let \(P_2\) denote the real vector space of polynomials of degree at most \(2\). An ordered basis \(\mathcal{B}=(b_1,b_2,b_3)\) of \(P_2\) is a list in which every polynomial in \(P_2\) has a unique expression as a linear combination of \(b_1,b_2,b_3\). The coordinate column \([p]_{\mathcal{B}}\) records those three coefficients in that order. For a linear map \(T:P_2\to P_2\), the notation \([T]_{\mathcal{C}\leftarrow\mathcal{B}}\) means that inputs are written in the domain basis \(\mathcal{B}\), while outputs are written in the codomain basis \(\mathcal{C}\).

The arrow in the subscript is a useful reminder: \(\mathcal{B}\)-coordinates go in, and \(\mathcal{C}\)-coordinates come out. Once a vector has been expressed in the domain basis, the matrix-vector product performs the map in coordinates.

Proof. Write \(\mathcal{B}=(b_1,\dots,b_n)\) and \([\mathbf{v}]_{\mathcal{B}}=(x_1,\dots,x_n)^{\top}\). By the definition of coordinates, \(\mathbf{v}=\sum_{j=1}^{n}x_jb_j\). Linearity gives \(T(\mathbf{v})=\sum_{j=1}^{n}x_jT(b_j)\). Taking \(\mathcal{C}\)-coordinates preserves sums and scalar multiples, so \([T(\mathbf{v})]_{\mathcal{C}}=\sum_{j=1}^{n}x_j[T(b_j)]_{\mathcal{C}}\). The columns of \([T]_{\mathcal{C}\leftarrow\mathcal{B}}\) are precisely these coordinate columns, and multiplying that matrix by \((x_1,\dots,x_n)^{\top}\) forms their indicated linear combination. This is the required equality. \(\square\)

Building and Applying a Mixed-Basis Matrix

Composing Maps: Match the Intermediate Coordinates

For a composition \(S\circ T\), the output coordinates of the matrix for \(T\) must be coordinates in the basis used for the input of the matrix for \(S\). The shared intermediate basis is what makes ordinary matrix multiplication represent the composition.

Proof. For any \(\mathbf{v}\in V\), the coordinate formula already proved gives \([T(\mathbf{v})]_{\mathcal{C}}=[T]_{\mathcal{C}\leftarrow\mathcal{B}}[\mathbf{v}]_{\mathcal{B}}\). Apply that formula to \(S\) with input \(T(\mathbf{v})\): \([S(T(\mathbf{v}))]_{\mathcal{D}}=[S]_{\mathcal{D}\leftarrow\mathcal{C}}[T(\mathbf{v})]_{\mathcal{C}}\). Substituting the first equality produces the product on the right side of the theorem, times \([\mathbf{v}]_{\mathcal{B}}\). Since this holds for every input coordinate column, the product has as its \(j\)-th column the \(\mathcal{D}\)-coordinates of \(S(T(b_j))\). Those are exactly the columns defining \([S\circ T]_{\mathcal{D}\leftarrow\mathcal{B}}\). \(\square\)

Adding Maps with the Same Coordinate Conventions

A sum \(T+U\) is defined when \(T\) and \(U\) have the same domain and codomain. To add their matrices entry by entry, use the same domain basis and the same codomain basis for both. This is not just a convenient convention: for every domain basis vector, the image under \(T+U\) is the sum of the two images, so its codomain coordinate column is the sum of their coordinate columns.

Proof. For each basis vector \(b_j\) of \(\mathcal{B}\), \((T+U)(b_j)=T(b_j)+U(b_j)\). Taking coordinates in the same basis \(\mathcal{C}\) gives \([(T+U)(b_j)]_{\mathcal{C}}=[T(b_j)]_{\mathcal{C}}+[U(b_j)]_{\mathcal{C}}\). The two terms on the right are the \(j\)-th columns of the matrices for \(T\) and \(U\). Thus every column of the matrix on the left is the corresponding column of the matrix sum, proving the equality. \(\square\)

Common Pitfall: Multiplying Matrices with the Wrong Middle Basis

It is possible for two matrices to have compatible sizes and still be incompatible as representations. In the composition example, \([S]_{\mathcal{D}\leftarrow\mathcal{C}}\) accepts \(\mathcal{C}\)-coordinates. It cannot be replaced by the matrix of \(S\) using the standard basis for both input and output. That standard-basis matrix is \(\begin{bmatrix}0&1&0\\0&0&2\\1&0&0\end{bmatrix}\), since \(S(1)=x^2\), \(S(x)=1\), and \(S(x^2)=2x\).

A tempting but incorrect calculation is to multiply this standard-basis matrix by \([T]_{\mathcal{C}\leftarrow\mathcal{B}}\). The resulting third row would be \((3,1,2)\), not the correct \((2,0,0)\). The product is wrong because the columns from \(T\) are \(\mathcal{C}\)-coordinates, not standard-basis coordinates. The correction is to use \([S]_{\mathcal{D}\leftarrow\mathcal{C}}\), which accepts exactly the coordinates that \(T\) produces.

Before multiplying or adding represented maps, check the basis labels as well as the matrix sizes. For composition, the first map’s output basis must match the second map’s input basis. For a sum, both maps must be represented with the same input and output bases. These checks prevent coordinate errors that ordinary matrix arithmetic cannot detect on its own.