Change of Basis Vectors for a Vector

1. Any \(N\)-Dimensional Vector represented in terms of Standard Basis Vectors can always be represented in terms of Any Set of \(N\) Arbitrary Basis Vectors each of \(N\)-Dimensions and vice versa as demonstrated in the following
   
   Lets consider a \(N\)-Dimensional Vector \(A\) having Components \(A_1, A_2, \cdots, A_n\) when represented using Standard Basis Vectors as follows
   
   \(\Rightarrow A=A_1\begin{bmatrix}1\\0\\ \vdots \\ 0_n\end{bmatrix} + A_2\begin{bmatrix}0\\1\\ \vdots \\ 0_n\end{bmatrix} + \cdots + A_n\begin{bmatrix}0\\0\\ \vdots \\ 1_n\end{bmatrix}\)    ...(1)
   
   \(\Rightarrow A=\begin{bmatrix}1 & 0& \cdots & 0_n \\ 0 & 1 & \cdots & 0_n \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1_n\end{bmatrix} \begin{bmatrix}A_1 \\ A_2 \\ \vdots \\A_n\end{bmatrix}\)   ...(2)
   
   The Vector \(A\) as given above can also be represented in terms of \(N\) Arbitrary \(N\)-Dimensional Basis Vectors \(e_1, e_2, \cdots , e_n\), having Components \(A_1', A_2', \cdots, A_n'\) as follows
   
   \(A = A_1'e_1 + A_2'e_2 + \cdots + A_n'e_n\)     where \(e_1=\begin{bmatrix}e_{11}\\e_{12}\\ \vdots \\e_{1n}\end{bmatrix},\hspace{.5cm}e_2=\begin{bmatrix}e_{21}\\e_{22}\\ \vdots \\e_{2n}\end{bmatrix},\hspace{.5cm}\cdots\hspace{.5cm}e_n=\begin{bmatrix}e_{n1}\\e_{n2}\\ \vdots \\e_{nn}\end{bmatrix}\)   ...(3)
   
   \(\Rightarrow A=A_1'\begin{bmatrix}e_{11}\\e_{12}\\ \vdots \\e_{1n}\end{bmatrix} + A_2'\begin{bmatrix}e_{21}\\e_{22}\\ \vdots \\e_{2n}\end{bmatrix} + \cdots + A_n'\begin{bmatrix}e_{n1}\\e_{n2}\\ \vdots \\e_{nn}\end{bmatrix}\)   ...(4)
   
   \(\Rightarrow A = \begin{bmatrix}e_{11} & e_{21} & \cdots & e_{n1} \\ e_{12} & e_{22} & \cdots & e_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ e_{1n} & e_{2n} & \cdots & e_{nn}\end{bmatrix}\begin{bmatrix}A'_1\\A'_2\\ \vdots \\ A'_n\end{bmatrix}\)   ...(5)
   
   Since equations (1) to (5) above all represent Vector \(A\), using equations (2) and (5) we get
   
   \(\begin{bmatrix}e_{11} & e_{21} & \cdots & e_{n1} \\ e_{12} & e_{22} & \cdots & e_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ e_{1n} & e_{2n} & \cdots & e_{nn}\end{bmatrix}\begin{bmatrix}A'_1\\A'_2\\ \vdots \\ A'_n\end{bmatrix}=\begin{bmatrix}1 & 0& \cdots & 0_n \\ 0 & 1 & \cdots & 0_n \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1_n\end{bmatrix} \begin{bmatrix}A_1 \\ A_2 \\ \vdots \\A_n\end{bmatrix}\)
   
   \(\Rightarrow \begin{bmatrix}e_{11} & e_{21} & \cdots & e_{n1} \\ e_{12} & e_{22} & \cdots & e_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ e_{1n} & e_{2n} & \cdots & e_{nn}\end{bmatrix}\begin{bmatrix}A'_1\\A'_2\\ \vdots \\ A'_n\end{bmatrix}=\begin{bmatrix}A_1 \\ A_2 \\ \vdots \\A_n\end{bmatrix}\)   ...(6)
   
   Multiplying equation (6) with the Inverse of the Matrix of the Arbitrary Basis Vector Set on both sides we get
   
   \(\begin{bmatrix}A'_1\\A'_2\\ \vdots \\ A'_n\end{bmatrix}=\begin{bmatrix}e_{11} & e_{21} & \cdots & e_{n1} \\ e_{12} & e_{22} & \cdots & e_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ e_{1n} & e_{2n} & \cdots & e_{nn}\end{bmatrix}^{-1}\begin{bmatrix}A_1\\A_2\\ \vdots \\A_n\end{bmatrix}\)   ...(7)
   
   The equation (6) aboves gives us the formula for Finding the Components of Vector \(A\) in terms of Standard Basis Vectors when the Vector \(A\) is represented in Set of Arbitrary Basis Vectors and it's Correspnding Components. Hence, To Represent any Vector in Standard Basis Vectors, the Vector represented in Set of Arbitrary Basis Vectors needs to be Pre-Multiplied By the Matrix of the Arbitrary Basis Vector Set.
   
   The equation (7) aboves gives us the formula for Finding the Components of Vector \(A\) in terms of Arbitrary Basis Vectors when the Vector \(A\) is given in terms of Standard Basis Vectors. Hence, To Represent any Vector in Any Set of Arbitrary Basis Vectors, the Vector represented in Standard Basis needs to be Pre-Multiplied By Inverse of the Matrix of the Arbitrary Basis Vector Set.
2. Any \(N\)-Dimensional Vector \(A\) represented using One Set of \(N\) Arbitrary Basis Vectors of \(N\)-Dimensions can be Represented using Any Other Set of \(N\) Arbitrary Basis Vectors \(N\)-Dimensions as demonstrated in the following
   
   Lets consider an \(N\)-Dimensional Vector \(A\) represented using One Set of \(N\) Arbitrary Basis Vectors of \(N\)-Dimensions \(e_1, e_2, ..., e_n\) through it's Corresponding Components \(A_1, A_2, \cdots, A_n\) and representated using Another Set of \(N\) Arbitrary Basis Vectors of \(N\)-Dimensions \(e_1', e_2', ..., e_n'\) through it's Corresponding Components \(A_1', A_2', \cdots, A_n'\) as follows
   
   \(A=A_1e_1 + A_2e_2 + ... + A_ne_n\)     where \(e_1=\begin{bmatrix}e_{11}\\e_{12}\\ \vdots \\e_{1n}\end{bmatrix},\hspace{.5cm}e_2=\begin{bmatrix}e_{21}\\e_{22}\\ \vdots \\e_{2n}\end{bmatrix},\hspace{.5cm}...\hspace{.5cm},\hspace{.5cm}e_n=\begin{bmatrix}e_{n1}\\e_{n2}\\ \vdots \\e_{nn}\end{bmatrix}\)   ...(8)
   
   Also, \(A=A'_1e_1' + A'_2e_2' + ... + A'_ne_n'\)     where \(e_1'=\begin{bmatrix}e'_{11}\\e'_{12}\\ \vdots \\e'_{1n}\end{bmatrix},\hspace{.5cm}e_2'=\begin{bmatrix}e'_{21}\\e'_{22}\\ \vdots \\e'_{2n}\end{bmatrix},\hspace{.5cm}...\hspace{.5cm},\hspace{.5cm}e_n'=\begin{bmatrix}e'_{n1}\\e'_{n2}\\ \vdots \\e'_{nn}\end{bmatrix}\)   ...(9)
   
   Since both equations (8) and (9) above represent Vector \(A\), we get
   
   \(A'_1e_1' + A'_2e_2' + ... + A'_ne_n'=A_1e_1 + A_2e_2 + ... + A_ne_n\)   ...(8)
   
   \(\Rightarrow A'_1\begin{bmatrix}e'_{11}\\e'_{12}\\ \vdots \\e'_{1n}\end{bmatrix} + A'_2\begin{bmatrix}e'_{21}\\e'_{22}\\ \vdots \\e'_{2n}\end{bmatrix} + ... + A'_n\begin{bmatrix}e'_{n1}\\e'_{n2}\\ \vdots \\e'_{nn}\end{bmatrix}=A_1\begin{bmatrix}e_{11}\\e_{12}\\ \vdots \\e_{1n}\end{bmatrix} + A_2\begin{bmatrix}e_{21}\\e_{22}\\ \vdots \\e_{2n}\end{bmatrix} + ... + A_n\begin{bmatrix}e_{n1}\\e_{n2}\\ \vdots \\e_{nn}\end{bmatrix}\)   ...(9)
   
   \(\Rightarrow \begin{bmatrix}e'_{11} & e'_{21} & \cdots & e'_{n1}\\ e'_{12} & e'_{22} & \cdots & e'_{n2}\\\vdots & \vdots & \ddots & \vdots \\ e'_{1n} & e'_{2n} & \cdots & e'_{nn} \end{bmatrix}\begin{bmatrix}A'_1\\A'_2\\ \vdots \\A'_n\end{bmatrix}=\begin{bmatrix}e_{11} & e_{21} & \cdots & e_{n1}\\ e_{12} & e_{22} & \cdots & e_{n2}\\\vdots & \vdots & \ddots & \vdots \\ e_{1n} & e_{2n} & \cdots & e_{nn} \end{bmatrix}\begin{bmatrix}A_1\\A_2\\ \vdots \\A_n\end{bmatrix}\)   ...(10)
   
   \(\Rightarrow \begin{bmatrix}A'_1\\A'_2\\ \vdots \\A'_n\end{bmatrix}=\begin{bmatrix}e'_{11} & e'_{21} & \cdots & e'_{n1}\\ e'_{12} & e'_{22} & \cdots & e'_{n2}\\\vdots & \vdots & \ddots & \vdots \\ e'_{1n} & e'_{2n} & \cdots & e'_{nn} \end{bmatrix}^{-1}\begin{bmatrix}e_{11} & e_{21} & \cdots & e_{n1}\\ e_{12} & e_{22} & \cdots & e_{n2}\\\vdots & \vdots & \ddots & \vdots \\ e_{1n} & e_{2n} & \cdots & e_{nn} \end{bmatrix}\begin{bmatrix}A_1\\A_2\\ \vdots \\A_n\end{bmatrix}\)   ...(11)
   
   \(\Rightarrow \begin{bmatrix}e_{11} & e_{21} & \cdots & e_{n1}\\ e_{12} & e_{22} & \cdots & e_{n2}\\\vdots & \vdots & \ddots & \vdots \\ e_{1n} & e_{2n} & \cdots & e_{nn} \end{bmatrix}^{-1}\begin{bmatrix}e'_{11} & e'_{21} & \cdots & e'_{n1}\\ e'_{12} & e'_{22} & \cdots & e'_{n2}\\\vdots & \vdots & \ddots & \vdots \\ e'_{1n} & e'_{2n} & \cdots & e'_{nn} \end{bmatrix}\begin{bmatrix}A'_1\\A'_2\\ \vdots \\A'_n\end{bmatrix}=\begin{bmatrix}A_1\\A_2\\ \vdots \\A_n\end{bmatrix}\)   ...(12)
   
   The equation (11) aboves gives the formula for Finding the Components of Vector \(A\) corresponding to Basis Vectors \(e_1', e_2', ..., e_n'\) when Components of Vector \(A\) corresponding to Basis Vectors \(e_1, e_2, ..., e_n\) are given.
   
   The equation (12) aboves gives the formula for Finding the Components of Vector \(A\) corresponding to Basis Vectors \(e_1, e_2, ..., e_n\) when Components of Vector \(A\) corresponding to Basis Vectors \(e_1', e_2', ..., e_n'\) are given.
3. In general, any Vector represented in terms of One Set of Basis Vectors (Source Basis Vector Set) can be represented in terms of Any Other Set of Basis Vectors (Target Basis Vector Set) under following conditions
   1. The Dimension of Basis Vector Matrix formed by the Source Basis Vector Set (also called the Source Basis Vector Matrix) must be same as the Dimension of Basis Vector Matrix formed by the Target Basis Vector Set (also called the Target Basis Vector Matrix).
   2. The Source Basis Vector Matrix must represent the either the Same Vector Space as the Target Basis Vector Matrix or a Vector Subspace of the Target Basis Vector Matrix.
   3. The given Vector must belong to the Column Space of both the Source and Target the Basis Vector Matrices.
   Provided the conditions given above are met, the following gives the formula for Finding the Components of a Vector in terms of Target Basis Vectors when the Components of that Vector are given in terms of Source Basis Vectors
   
   \(X'={T_D}^{T}SX\)   ...(13)
   
   where
   
   \(X=\) The Column Matrix containing the Components of Vector in terms of Source Basis Vectors
   \(X'=\) The Column Matrix containing the Components of Vector in terms of Target Basis Vectors
   \(S=\) The Matrix Containing Source Basis Vectors corresponding to \(X\) (i.e. the Source Basis Vector Matrix)
   \(T=\) The Matrix Containing Target Basis Vectors corresponding to \(X'\) (i.e. the Target Basis Vector Matrix)
   \(T_D=\) Dual Matrix of the Target Basis Vector Matrix
   
   If Any One of the Target Basis Vectors is a Complex Vector (which means the Target Basis Vector Matrix is a Complex Matrix), then the Components of the Vector in terms of Target Basis Vectors can also be found using the following formula
   
   \(X'={T_D}^{\dagger}SX\)   ...(14)
   
   If the Target Basis Vector Matrix is a Square Matrix, then the formuale given in equation (13) and (14) get reduced to
   
   \(X'=T^{-1}SX\)   ...(15)
   
   The Matrices \({T_D}^{T}S\) , \({T_D}^{\dagger}S\) and \(T^{-1}S\) in equations (13), (14) and (15) respectively are called the Vector Transformation Matrices, which are used to Transform any Vector's Representation from Set of Source Basis Vectors to Set of Target Basis Vectors.