Constructing a Change-of-Basis Matrix

From One Coordinate Description to Another

A vector does not change when we describe it using a different basis, but its coordinate vector usually does. A change-of-basis matrix records how to convert between those descriptions. Its direction matters: here, the input is a coordinate vector relative to \(\mathcal{B}\), and the output is the coordinate vector relative to \(\mathcal{C}\).

Throughout, \(V\) is a finite-dimensional vector space over \(\mathbb{R}\), and \(\mathcal{B}=(\mathbf{b}_1,\dots,\mathbf{b}_n)\) and \(\mathcal{C}=(\mathbf{c}_1,\dots,\mathbf{c}_n)\) are ordered bases of \(V\). For a vector \(\mathbf{v}\in V\), its coordinate vector \([\mathbf{v}]_{\mathcal{B}}\) is the column of coefficients in its unique basis expansion \(\mathbf{v}=\alpha_1\mathbf{b}_1+\cdots+\alpha_n\mathbf{b}_n\); thus \([\mathbf{v}]_{\mathcal{B}}=(\alpha_1,\dots,\alpha_n)^{\top}\). The order of the basis vectors determines the order of the entries in this column.

To find one column, take a basis vector \(\mathbf{b}_i\) and solve \(\mathbf{b}_i=x_1\mathbf{c}_1+\cdots+x_n\mathbf{c}_n\). The solution column \((x_1,\dots,x_n)^{\top}\) gives the coefficients needed to express \(\mathbf{b}_i\) in the \(\mathcal{C}\) basis. Since \(\mathcal{C}\) is a basis, these coefficients exist and are unique. Repeating this for each \(\mathbf{b}_i\) constructs the matrix.

Why These Columns Convert Coordinates

Proof. Let \(\mathbf{v}\in V\). Because \(\mathcal{B}\) is a basis, there are unique scalars \(\alpha_1,\dots,\alpha_n\) such that \(\mathbf{v}=\sum_{i=1}^{n}\alpha_i\mathbf{b}_i\), and the column of these scalars is \([\mathbf{v}]_{\mathcal{B}}\). By the definition of the columns of \(\mathbf{P}_{\mathcal{C}\leftarrow\mathcal{B}}\), the \(i\)th column is \([\mathbf{b}_i]_{\mathcal{C}}\). Matrix multiplication therefore gives \(\mathbf{P}_{\mathcal{C}\leftarrow\mathcal{B}}[\mathbf{v}]_{\mathcal{B}}=\sum_{i=1}^{n}\alpha_i[\mathbf{b}_i]_{\mathcal{C}}\). Each column on the right lists the coefficients of \(\mathbf{b}_i\) in the \(\mathcal{C}\) basis. Substituting those expansions into \(\sum_i\alpha_i\mathbf{b}_i=\mathbf{v}\) shows that the resulting coefficients are precisely the coefficients of \(\mathbf{v}\) in that basis. Since \(\mathcal{C}\)-coordinates are unique, the column is \([\mathbf{v}]_{\mathcal{C}}\), as required. \(\square\)

This result explains why the columns are placed in that order: the first input coordinate multiplies the first column, the second input coordinate multiplies the second column, and so on. It also provides a reliable construction procedure. Find the coordinates of each vector in the starting basis \(\mathcal{B}\), using the target basis \(\mathcal{C}\), and place the answers as columns.

Finding the Columns by Solving Systems

When several coordinate equations share the same target basis, one row reduction can solve them together. The coefficient matrix has the target basis vectors as columns, and the right-hand side has the starting basis vectors as columns. Row-reducing the augmented matrix then produces all the desired coordinate columns at once.

A General Construction and Its Conditions

The preceding procedure works because each vector of \(V\) has exactly one coordinate column relative to a basis. More formally, the equation for the \(i\)th column is \(\mathbf{b}_i=\sum_{j=1}^{n}p_{ji}\mathbf{c}_j\), where \(p_{ji}\) is the entry in row \(j\), column \(i\) of the transition matrix. The column index \(i\) tells us which \(\mathcal{B}\) vector is being expressed; the row index \(j\) tells us the coefficient of \(\mathbf{c}_j\).

Proof of the construction rule. Since \(\mathcal{C}\) spans \(V\), each \(\mathbf{b}_i\in V\) can be written as a linear combination of the \(\mathbf{c}_j\). Since \(\mathcal{C}\) is linearly independent, this expression is unique. Thus the equation determines one and only one column for every \(i\), giving a well-defined matrix. The theorem above then shows that this matrix converts the coordinates of every vector, not just the basis vectors used to build it. \(\square\)

In \(\mathbb{R}^n\), if the basis vectors are given in standard coordinates, the same construction can be written as a system. Let \(\mathbf{C}\) be the matrix whose columns are the vectors of \(\mathcal{C}\), and let \(\mathbf{B}\) have the vectors of \(\mathcal{B}\) as columns. The coordinate equations for all the columns together are \(\mathbf{C}\mathbf{P}_{\mathcal{C}\leftarrow\mathcal{B}}=\mathbf{B}\). This is a compact way to organize the systems, not a different definition: each column equation is still the expansion of one \(\mathcal{B}\) vector in the target basis.

For this system, \(\mathbf{C}\) must be square and its columns must be a basis of \(\mathbb{R}^n\). If the target list does not span the space, some starting vectors may have no expression in it. If it is not linearly independent, a vector may have more than one expression. In either case, it does not define a coordinate system for all of \(V\), and the construction cannot be used as stated.

Common Mistake: Reversing the Direction

The notation \(\mathbf{P}_{\mathcal{C}\leftarrow\mathcal{B}}\) means that \(\mathcal{B}\)-coordinates go in and \(\mathcal{C}\)-coordinates come out. A common error is to put the coordinate vectors of \(\mathcal{C}\) in the \(\mathcal{B}\) basis into the columns, or to transpose the correct matrix. That produces a different operation.

For instance, use the matrix from the first example and input \((2,-1)^{\top}\). The correct calculation is \(\mathbf{P}_{\mathcal{C}\leftarrow\mathcal{B}}(2,-1)^{\top}=(\tfrac52,\tfrac52)^{\top}\). A mistaken transpose gives \(\begin{bmatrix}2&1\\ \tfrac32&-\tfrac12\end{bmatrix}(2,-1)^{\top} =(3,\tfrac72)^{\top}\). Reconstructing from that incorrect output gives \(3(1,1)^{\top}+\tfrac72(1,-1)^{\top}=(\tfrac{13}{2},-\tfrac12)^{\top}\), not the original vector \((5,0)^{\top}\). The check exposes the error. To correct it, place \([\mathbf{b}_1]_{\mathcal{C}}\) and \([\mathbf{b}_2]_{\mathcal{C}}\) in the first and second columns, respectively, and verify that the resulting output reconstructs the input vector in the target basis.

A useful final check is therefore to apply the proposed matrix to at least one coordinate vector and reconstruct the resulting vector using the target basis. For a complete check of the construction, multiply the target basis matrix by the proposed transition matrix: the product must have the starting basis vectors as its columns. This confirms both the entries and the direction.