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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 \(P \times Q\) Matrix \(L\) (which is 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. If Both Basis Vector Matrices \(S\) and \(T\) represent the Same Vector Space/Subspace then
   1. The Basis Vector Transformation Matrix \(L\) is a Square Matrix.
   2. The Target Basis Vector Matrix \(T\) can be converted back to Source Basis Vector Matrix \(S\) by Post Mutiplying it with Inverse of Basis Vector Transformation Matrix \(L\) (i.e. \(L^{-1}\)). That is
      
      \(S = TL^{-1}\)   ...(9)
      
      The Matrix \(L^{-1}\) can also be calculated as
      
      \(L^{-1} = {T_D}^T S\)   ...(10) (where \(T_D\) is Dual Matrix of \(T\))
      
      If the Target Basis Vector Matrix \(T\) is a Complex Matrix, then \(L^{-1}\) can also be calculated as
      
      \(L^{-1} = {T_D}^{\dagger}S\)   ...(11)
      
      If the Target Basis Vector Matrix \(T\) is a Square Matrix, then the formuale given in equation (10) and (11) get reduced to
      
      \(L^{-1} = T^{-1}S\)   ...(12)
   3. The Basis Vector Transformation Matrix that Transforms the Basis Vector Matrix \(S\) to \(T\) is an Inverse of Vector Component Transformation Matrix that Transforms the Vector Components given in Basis Vector Matrix \(S\) to Vector Components given in Basis Vector Matrix \(T\).
      
      Likewize, the Basis Vector Transformation Matrix that Transforms the Basis Vector Matrix \(T\) to \(S\) is an Inverse of Vector Component Transformation Matrix that Transforms the Vector Components given in Basis Vector Matrix \(T\) to Vector Components given in Basis Vector Matrix \(S\).
4. You can use the Basis Vector Transformation Matrix Calculator to calculate Basis Vector Transformation Matrix.