Coordinate System and Change of Basis

Let \(\mathcal{B} = \{b_1, b_2, ..., b_n\}\) be a basis for a vector space \(V\). Then for \(x = [x_1, x_2, ..., x_n]^T\) in \(V\), there exists a unique set of scalars \(q_1, q_2, ..., p_n\) such that: \[ x = q_1b_1 + q_2b_2 + ... + q_nb_n \] These scalars are called the coordinates of \(x\) relative to the basis \(\mathcal{B}\). \[ [x]_\mathcal{B} = \begin{bmatrix} q_1 \cdots q_n \end{bmatrix}^T \] is the coordinate vectors of \(x\) relative to \(\mathcal{B}\). The mapping \(x \mapsto [x]_\mathcal{B}\) is called the coordinate mapping determined by \(\mathcal{B}\).

Let \(P_\mathcal{B} = [b_1, b_2, ..., b_n]\), then \(x = P_\mathcal{B}[x]_\mathcal{B}\)

Let \(\mathcal{B}\) and \(\mathcal{C}\) both be a basis for an n-dimensional vector space \(\R^n\). Then there is a unique \(n \times n\) matrix \(\mathop{P}\limits_\mathcal{C \leftarrow B}\) such that: \[ [x]_\mathcal{C} = \mathop{P}\limits_\mathcal{C \leftarrow B}[x]_\mathcal{B} \] The columns of \(\mathop{P}\limits_\mathcal{C \leftarrow B}\) are the \(\mathcal{C}\)-coordinate vectors of the vectors in the basis \(\mathcal{B}\): \[ \mathop{P}\limits_\mathcal{C \leftarrow B} = [[b_1]_\mathcal{C}, [b_2]_\mathcal{C}, ..., [b_n]_\mathcal{C}] \]

\[ \begin{aligned} &P_\mathcal{C}\mathop{P}\limits_\mathcal{C \leftarrow B}[x]_\mathcal{B} = P_\mathcal{C}\mathop{P}\limits_\mathcal{C \leftarrow B} \begin{bmatrix} q_1 \cdots q_n \end{bmatrix}^T \\ &= P_\mathcal{C}(q_1[b_1]_\mathcal{C} + q_2[b_2]_\mathcal{C} + ... + q_n[b_n]_\mathcal{C}) \\ &= q_1P_\mathcal{C}[b_1]_\mathcal{C} + q_2P_\mathcal{C}[b_2]_\mathcal{C} + ... + q_nP_\mathcal{C}[b_n]_\mathcal{C} \\ &= q_1b_1 + q_2b_2 + ... + q_nb_n \\ &= x \end{aligned} \]

Specifically, when \(\mathcal{B} = \mathcal{I}\) is the standard basis, \[ \begin{aligned} \mathop{P}\limits_\mathcal{C \leftarrow I} &= [[e_1]_\mathcal{C}, [e_2]_\mathcal{C}, ..., [e_n]_\mathcal{C}] \\ P_\mathcal{C}\mathop{P}\limits_\mathcal{C \leftarrow I} &= P_\mathcal{C}[[e_1]_\mathcal{C}, [e_2]_\mathcal{C}, ..., [e_n]_\mathcal{C}] \\ &= [e_1, e_2, ..., e_n] \\ &= I \end{aligned} \] Since \(P_\mathcal{C}\) is invertible, \(\mathop{P}\limits_\mathcal{C \leftarrow I} = P_\mathcal{C}^{-1}\). Therefore, \([x]_\mathcal{B} = P_\mathcal{B}^{-1}x\) in equation (2)

\[ \begin{aligned} \\ [x]_\mathcal{C} &= \mathop{P}\limits_\mathcal{C \leftarrow B}[x]_\mathcal{B} \\ (\mathop{P}\limits_\mathcal{C \leftarrow B})^{-1}[x]_\mathcal{C} &= (\mathop{P}\limits_\mathcal{C \leftarrow B})^{-1}\mathop{P}\limits_\mathcal{C \leftarrow B}[x]_\mathcal{B} \\ [x]_\mathcal{B} &= (\mathop{P}\limits_\mathcal{C \leftarrow B})^{-1}[x]_\mathcal{C} \\ \end{aligned} \] In other words, \(\mathop{P}\limits_\mathcal{B \leftarrow C} = (\mathop{P}\limits_\mathcal{C \leftarrow B})^{-1}, \mathop{P}\limits_\mathcal{B \leftarrow C}\mathop{P}\limits_\mathcal{C \leftarrow B} = I\)