Subsection 2.1.4 Linear Transformations
The transformation in which the components of a vector in one coordinate system are linear functions of another coordinate system is called a linear transformation. In sych transformation origin of two coordinate systems do not displace. If \(x_{i}\) are the components of a vector in one coordinate system and \(y_{i}\) are those in another coordinate system then after a linear transformation, we have -
\begin{equation*}
y_{i}=b_{i1}x_{1}+b_{i2}x_{2}+\cdots+b_{in}x_{n}
\end{equation*}
\begin{equation}
= \sum\limits_{k=1}^{n}b_{ik}x_{k}, \quad 1 \leq i\leq n.\tag{2.1.6}
\end{equation}
Again, the same vector has components \(z_{i}\text{,}\) which are linearly related to the components \(y_{i}\) then after a linear transformation, we have -
\begin{equation}
z_{i}= \sum\limits_{k=1}^{n}a_{ik}y_{k}, \quad 1 \leq i\leq n.\tag{2.1.7}
\end{equation}
Now it is possible to obtain a transformation directly from the components \(x_{i}\text{,}\) from eqns. (2.1.6) and (2.1.7),
\begin{equation*}
z_{i}= \sum\limits_{k=1}^{n}a_{ik} \sum\limits_{j=1}^{n}b_{kj}x_{j}
\end{equation*}
\begin{equation}
= \sum\limits_{j=1}^{n} \sum\limits_{k=1}^{n} a_{ik} b_{kj}x_{j} = \sum\limits_{j=1}^{n}c_{ij}x_{j}, \quad 1 \leq i\leq n.\tag{2.1.8}
\end{equation}
where, \(c_{ij} = \sum\limits_{k=1}^{n}a_{ik}b_{kj}, \quad 1 \leq i, j\leq n.\)
