Common Basis-Order and Coordinate-Convention Errors

Read the Basis Labels Before the Entries

A matrix representing a linear map is not just a rectangular array attached to that map. Its entries depend on the chosen ordered basis of the domain and the chosen ordered basis of the codomain. The labels \([T]_{\mathcal{C}\leftarrow\mathcal{B}}\) record those choices: the matrix accepts coordinates in \(\mathcal{B}\) and returns coordinates in \(\mathcal{C}\). Many errors that look like arithmetic mistakes are actually failures to track this input and output convention.

Throughout this tutorial, the spaces are finite-dimensional over \(\mathbb{R}\). An ordered basis \(\mathcal{B}=(\mathbf{b}_1,\dots,\mathbf{b}_n)\) of a vector space \(V\) gives each \(\mathbf{v}\in V\) a unique coordinate vector \([\mathbf{v}]_{\mathcal{B}}\in\mathbb{R}^n\), defined by \(\mathbf{v}=x_1\mathbf{b}_1+\cdots+x_n\mathbf{b}_n\) when \([\mathbf{v}]_{\mathcal{B}}=(x_1,\dots,x_n)^{\top}\). The order of the basis vectors matters because it determines the order of the coordinates.

Keep the two roles separate. The basis to the right of the arrow is the domain basis and determines the input coordinates. The basis to the left is the codomain basis and determines how the output is expressed. If \(T:V\to W\), then the number of columns is \(\dim V\) and the number of rows is \(\dim W\). A matrix with the right dimensions can still be wrong if either basis label is wrong.

Why Columns, Not Rows, Record the Images

Proof. Let \(\mathbf{A}=[T]_{\mathcal{C}\leftarrow\mathcal{B}}\), and let \(\mathbf{a}_j\) be its \(j\)th column. The coordinate vector of \(\mathbf{b}_j\) in the ordered basis \(\mathcal{B}\) is the \(j\)th standard coordinate vector \(\mathbf{e}_j\). By the defining property of the representation matrix, \([T(\mathbf{b}_j)]_{\mathcal{C}}=\mathbf{A}\mathbf{e}_j\). Multiplication by \(\mathbf{e}_j\) selects column \(j\), so \(\mathbf{a}_j=[T(\mathbf{b}_j)]_{\mathcal{C}}\). This argument applies to every \(j\) from \(1\) to \(n\). \(\square\)

This is why the standard construction places the coordinate vectors of the images of the domain basis vectors in the columns. Rows have a different role: a row collects the contributions to one output coordinate from all input coordinates. Reading a row as the image of a basis vector confuses these two roles.

Swapping Bases Changes What the Coordinates Mean

Even for the same map, changing the domain basis or codomain basis generally changes the matrix. The identity map makes this dependence especially visible: it leaves each vector unchanged, but its input and output coordinates can be different when the bases differ. In particular, \([T]_{\mathcal{B}\leftarrow\mathcal{C}}\) is not an instruction to take \(\mathcal{B}\)-coordinates as input. Its input is \(\mathcal{C}\)-coordinates, and its output is \(\mathcal{B}\)-coordinates.

Composition Matrices Multiply in Application Order

Suppose \(S:U\to V\) is applied first and \(T:V\to W\) second. The output coordinates of \(S\) must be the input coordinates for \(T\). Thus the basis used to express the intermediate vector must agree on both sides of the matrix product. This agreement is what makes the order of multiplication unavoidable.

Proof. Let \(\mathbf{u}\in U\). By the defining property of the matrix of \(S\), \([S(\mathbf{u})]_{\mathcal{B}}=[S]_{\mathcal{B}\leftarrow\mathcal{A}}[\mathbf{u}]_{\mathcal{A}}\). Apply the defining property of the matrix of \(T\) to the vector \(S(\mathbf{u})\in V\): \([T(S(\mathbf{u}))]_{\mathcal{C}}=[T]_{\mathcal{C}\leftarrow\mathcal{B}}[S(\mathbf{u})]_{\mathcal{B}}\). Substituting the first equality into the second gives \[ [T(S(\mathbf{u}))]_{\mathcal{C}} = [T]_{\mathcal{C}\leftarrow\mathcal{B}} [S]_{\mathcal{B}\leftarrow\mathcal{A}} [\mathbf{u}]_{\mathcal{A}}. \] This holds for every \(\mathbf{u}\in U\), so the product matrix represents \(T\circ S\) from \(\mathcal{A}\)-coordinates to \(\mathcal{C}\)-coordinates. \(\square\)

The rightmost matrix acts first. In \([T]_{\mathcal{C}\leftarrow\mathcal{B}}[S]_{\mathcal{B}\leftarrow\mathcal{A}}\), the right factor takes \(\mathcal{A}\)-coordinates to \(\mathcal{B}\)-coordinates; the left factor then takes \(\mathcal{B}\)-coordinates to \(\mathcal{C}\)-coordinates. Reversing the factors generally fails both the basis-label check and the numerical check.

A Reliable Convention Check

Before trusting a proposed matrix, check its labels and its action. These checks are short, but they catch different errors:

• Check dimensions. If the domain has dimension \(n\) and the codomain has dimension \(m\), the matrix must have \(n\) columns and \(m\) rows.
• Check a basis vector. Column \(j\) should equal the codomain-coordinate vector of the image of domain basis vector \(j\).
• Check coordinate meanings. The input column must use the basis named on the right of the arrow, and the output must be interpreted in the basis named on the left.
• Check composition compatibility. The output basis of the right-hand factor must be the input basis of the left-hand factor. The right-hand factor acts first.

When diagnosing a wrong answer, ask whether it fails because of a numerical computation or because the coordinate vector was assigned the wrong meaning. For example, the multiplication in the reversed-basis example is numerically correct: the error occurs only when \(\mathcal{B}\)-coordinates are supplied where \(\mathcal{C}\)-coordinates are required. Likewise, a transposed matrix may have the correct entries but put image vectors in rows rather than columns. Stating the bases beside every matrix makes these distinctions visible.