Intuitive understanding of multiplying a matrix
\[A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \quad x=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \quad \hat i=\begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad \hat j=\begin{bmatrix} 0 \\ 1 \end{bmatrix}\] \[x=x_1\begin{bmatrix}1 \\ 0\end{bmatrix} + x_2\begin{bmatrix}0 \\ 1 \end{bmatrix}\] \[Ax = x_1 \times A\begin{bmatrix}1 \\ 0\end{bmatrix} + x_2 \times A\begin{bmatrix}0 \\ 1\end{bmatrix} = x_1\begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} + x_2\begin{bmatrix} a_{12} \\ a_{22} \end{bmatrix}\]Multiplying matrices can be seen as change of basis
\[A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \quad B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \quad\] \[BAI = BA = B \times \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \left[ \begin{array}{c|c} B \begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} & B \begin{bmatrix} a_{12} \\ a_{22} \end{bmatrix} \end{array}\right]\] \[BAx = x_1 \times B \begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} + x_2 \times B \begin{bmatrix} a_{12} \\ a_{22} \end{bmatrix}\]Applying matrices several time can be seen as applying linear transformations several time
\[A \in \mathbb{R}^{n\times m} \quad x \in \mathbb{R}^m\]If \(A\) is non-squared, it transforms a vector to another dimension.
if \(n < m\), then \(A\) squeezes \(m\)-dimensional vector in to \(n\)-dimensinal vector
if \(n = 1\) and \(m = 3\), multiplying \(A\) squeezes \(3\)-dimensional vector into scalar. Generally, in this case, a plane squeezes into a scalar.
else if \(n > m\), \(A\) puts \(m\)-dimensional space in \(n\)-dimensional space
if \(n = 3\) and \(m = 1\), generally, multiplying \(A\) puts a scalar into \(3\)-dimensional space.
