Linear Algebra@Chapter 2

Multiplication of Matrix

First at all, understand $\begin{bmatrix}2 & 3\\ 4 & 5\end{bmatrix} \begin{bmatrix}1\\ 2\end{bmatrix}=1\begin{bmatrix}2\\ 4\end{bmatrix}+2\begin{bmatrix}3\\ 5\end{bmatrix}$

Then, you can understand $AB=\begin{bmatrix}Ab_{1} &Ab_{2}... & Ab_{i}\end{bmatrix}$
only $Ab$ can sum to one column, $A$ is coefficient matrix, $b$ is the vector of variables, $Ab$ can sum to one column if the linear system is completed, i.e $cx_{1}+bx_{2}$. Different column in $B$ means different linear system for $A$, so $Ab_{1}$ will not sum with $Ab_{2}....Ab_{i}$.

$\begin{bmatrix}2 & 3\\ 4 & 5\end{bmatrix} \begin{bmatrix}1 &2 \\ 6&7\end{bmatrix}=\begin{bmatrix}2 & 3\\ 4 & 5\end{bmatrix} \begin{bmatrix}1\\ 6\end{bmatrix}+\begin{bmatrix}2 & 3\\ 4 & 5\end{bmatrix} \begin{bmatrix}2\\ 7\end{bmatrix}$
$\begin{bmatrix}2 & 3\\ 4 & 5\end{bmatrix} \begin{bmatrix}1\\ 6\end{bmatrix}=\begin{bmatrix}2 &18 \\ 4&30 \end{bmatrix}=\begin{bmatrix}20\\ 34\end{bmatrix}$
$\begin{bmatrix}2 & 3\\ 4 & 5\end{bmatrix} \begin{bmatrix}2\\ 7\end{bmatrix}=\begin{bmatrix}4 & 6\\ 8& 35\end{bmatrix}=\begin{bmatrix}10\\ 43\end{bmatrix}$
Finally,$\begin{bmatrix}20 & 10\\ 34 & 43\end{bmatrix}$

Transpose of matrix
$A=\begin{bmatrix}1 &1 &1 \\ 2&3 & 4\end{bmatrix}$

$A^{T}=\begin{bmatrix}1 &2 \\ 1 & 3\\1 & 4\end{bmatrix}$