Change of Coordinates Matrix — Definition, Formula & Examples

A change of coordinates matrix is a matrix that converts the coordinate representation of a vector in one basis to its coordinate representation in another basis. It tells you how to "translate" between two different coordinate systems for the same vector space.

Let $\mathcal{B} = \{\mathbf{b}_1, \ldots, \mathbf{b}_n\}$ and $\mathcal{C} = \{\mathbf{c}_1, \ldots, \mathbf{c}_n\}$ be two bases for a vector space $V$ . The change of coordinates matrix from $\mathcal{B}$ to $\mathcal{C}$ , denoted $P_{\mathcal{B} \leftarrow \mathcal{C}}$ or $P_{\mathcal{C} \to \mathcal{B}}$ , is the unique $n \times n$ invertible matrix such that $[\mathbf{x}]_{\mathcal{B}} = P_{\mathcal{B} \leftarrow \mathcal{C}} [\mathbf{x}]_{\mathcal{C}}$ for every $\mathbf{x} \in V$ . Its columns are the $\mathcal{B}$ -coordinate vectors of each basis vector in $\mathcal{C}$ .

Key Formula

[\mathbf{x}]_{\mathcal{B}} = P_{\mathcal{B} \leftarrow \mathcal{C}} \, [\mathbf{x}]_{\mathcal{C}}

Where:

$[\mathbf{x}]_{\mathcal{B}}$ = Coordinate vector of x with respect to basis B
$[\mathbf{x}]_{\mathcal{C}}$ = Coordinate vector of x with respect to basis C
$P_{\mathcal{B} \leftarrow \mathcal{C}}$ = Change of coordinates matrix from C-coordinates to B-coordinates

How It Works

To build the change of coordinates matrix from basis

\mathcal{C}

to basis

\mathcal{B}

, express each vector in

\mathcal{C}

as a linear combination of the vectors in

\mathcal{B}

. Place the resulting coordinate vectors as the columns of a matrix. To convert a vector's coordinates from

\mathcal{C}

\mathcal{B}

, multiply this matrix by the

\mathcal{C}

-coordinate vector. To go the other direction (from

\mathcal{B}

\mathcal{C}

), use the inverse of this matrix.

Worked Example

Problem: In R², let the standard basis be B = {e₁, e₂} and let C = {(1, 1), (1, −1)}. Find the change of coordinates matrix from C to B, then convert the vector x with C-coordinates [x]_C = (3, 2) into standard coordinates.

Step 1: Express each basis vector of C in terms of B (the standard basis). Since B is standard, the columns are simply the vectors of C themselves.

P_{\mathcal{B} \leftarrow \mathcal{C}} = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}

Step 2: Multiply the change of coordinates matrix by the C-coordinate vector.

[\mathbf{x}]_{\mathcal{B}} = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}

Answer: The standard coordinates of x are (5, 1). This means x = 3(1,1) + 2(1,−1) = (5, 1).

Why It Matters

Change of coordinates matrices are essential in diagonalization, where you convert a matrix into a diagonal form using eigenvector bases. They also appear in computer graphics for switching between world, camera, and screen coordinate systems, and in physics when transforming between reference frames.

Common Mistakes

Mistake: Reversing the direction of the matrix — using P from B to C when you need P from C to B.

Correction: Pay close attention to notation. The columns of P_{B←C} are the C-basis vectors expressed in B-coordinates. If you need the opposite direction, take the inverse.