Basis Vector Transformation

1. Basis Vector Transformation referes to converting a \(M \times P\) Source Basis Vector Matrix \(S\) to a \(M \times Q\) Target Basis Vector Matrix \(T\) (where \(2 \leq Q \leq P\)) by Post Mutiplying it with a \(P \times Q\) Basis Vector Matrix \(L\) (also called the Basis Vector Transformation Matrix). That is
   
   \(T = SL\)   ...(1)
   
   For any given Pair of Basis Vector Matrices \(S\) and \(T\), a Basis Vector Transformation Matrix \(L\) can exist only under following conditions
   1. The Dimension of Basis Vector Matrices \(S\) and \(T\) must be same.
   2. The Target Basis Vector Matrix \(T\) must represent the either the Same Vector Space as the Source Basis Vector Matrix \(S\) or a Vector Subspace of the Source Basis Vector Matrix \(S\) .
   Provided the conditions given above are met, the Basis Vector Transformation Matrix \(L\) can be calculated as
   
   \(L = {S_D}^T T\)   ...(2) (where \(S_D\) is Dual Matrix of \(S\))
   
   If the Source Basis Vector Matrix \(S\) is a Complex Matrix, then the Basis Vector Transformation Matrix \(L\) can also be calculated as
   
   \(L = {S_D}^{\dagger}T\)   ...(3)
   
   If the Source Basis Vector Matrix \(S\) is a Square Matrix, then the formuale given in equation (2) and (3) get reduced to
   
   \(L = S^{-1}T\)   ...(4)
2. Basis Vector Transformation can also refer to Representing Each Vector in One Set of Basis Vector (Target Basis Vectors) in terms of a Vector of Some Other Set of Basis Vector (Source Basis Vectors). For example, let's consider \(M\)-Dimensional Source Basis Vector Set \(S\) consisting of \(P\) Basis Vectors (\(\vec{e_1}, \vec{e_2}, ..., \vec{e_p}\)) and \(M\)-Dimensional Target Basis Vector Set \(T\) consisting of \(Q\) Basis Vectors (\(\vec{e'_1}, \vec{e'_2}, ..., \vec{e'_q}\)) (where \(2 \leq Q \leq P\)) as given below
   
   \(S=\begin{bmatrix}\vec{e_1} & \vec{e_2} & \cdots & \vec{e_p}\end{bmatrix}=\begin{bmatrix}e_{11} & e_{12} & \cdots & e_{1p}\\ e_{21} & e_{22} & \cdots & e_{2p}\\\vdots & \vdots & \ddots & \vdots \\ e_{m1} & e_{m2} & \cdots & e_{mp} \end{bmatrix}\hspace{9mm} T=\begin{bmatrix}\vec{e_1'} & \vec{e_2'} & \cdots & \vec{e_q'}\end{bmatrix}=\begin{bmatrix}e'_{11} & e'_{12} & \cdots & e'_{1q}\\ e'_{21} & e'_{22} & \cdots & e'_{2q}\\\vdots & \vdots & \ddots & \vdots \\ e'_{m1} & e'_{m2} & \cdots & e'_{mq} \end{bmatrix}\)
   
   Now, Basis Vector Transformation from \(S\) to \(T\) requires Each Vector in Basis Vector Set \(T\) to be Represented in terms of a Vector given in Basis Vector Set \(S\) as follows
   
   \(\vec{e_1'}=A_{11}\vec{e_1} + A_{21}\vec{e_2} + ... + A_{p1}\vec{e_p}\)
   \(\vec{e_2'}=A_{12}\vec{e_1} + A_{22}\vec{e_2} + ... + A_{p2}\vec{e_p}\)
   \(\vdots\)
   \(\vec{e_q'}=A_{1q}\vec{e_1} + A_{2q}\vec{e_2} + ... + A_{pq}\vec{e_p}\)   ...(5)
   
   The above Transformation can be Represented in form of a Matrix Product Equation as follows
   
   \(\begin{bmatrix}\vec{e_1'} & \vec{e_2'} & \cdots & \vec{e_q'}\end{bmatrix}=\begin{bmatrix}\vec{e_1} & \vec{e_2} & \cdots & \vec{e_p}\end{bmatrix}\begin{bmatrix}A_{11} & A_{12} & \cdots & A_{1q}\\ A_{21} & A_{22} & \cdots & A_{2q}\\\vdots & \vdots & \ddots & \vdots\\A_{p1} & A_{p2} & \cdots & A_{pq}\end{bmatrix}\)   ...(6)
   
   \(\Rightarrow \begin{bmatrix}e'_{11} & e'_{12} & \cdots & e'_{1q}\\ e'_{21} & e'_{22} & \cdots & e'_{2q}\\\vdots & \vdots & \ddots & \vdots \\ e'_{m1} & e'_{m2} & \cdots & e'_{mq} \end{bmatrix}=\begin{bmatrix}e_{11} & e_{12} & \cdots & e_{1p}\\ e_{21} & e_{22} & \cdots & e_{2p}\\\vdots & \vdots & \ddots & \vdots \\ e_{m1} & e_{m2} & \cdots & e_{mp} \end{bmatrix}\begin{bmatrix}A_{11} & A_{12} & \cdots & A_{1q}\\ A_{21} & A_{22} & \cdots & A_{2q}\\\vdots & \vdots & \ddots & \vdots\\A_{p1} & A_{p2} & \cdots & A_{pq}\end{bmatrix}\)   ...(7)
   
   \(\Rightarrow T=SL\)   ...(8 )
   
   where Basis Vector Transformation Matrix \(L=\begin{bmatrix}A_{11} & A_{12} & \cdots & A_{1q}\\ A_{21} & A_{22} & \cdots & A_{2q}\\\vdots & \vdots & \ddots & \vdots\\A_{p1} & A_{p2} & \cdots & A_{pq}\end{bmatrix}\) can be calculated as given in equations (2), (3) and (4) above.
3. You can use the Basis Vector Transformation Matrix Calculator to calculate Basis Vector Transformation Matrix.