How to view matrix multiplication?
\[ \begin{bmatrix} 2 & 1 & 3 \\ 3 & 1 & 4 \\ 5 & 7 & 12 \end{bmatrix}{\cdot} \begin{bmatrix} x1 \\ x2 \\ x3 \end{bmatrix} \]
Instead of viewing it as a dot product, the output should be seen as a combination of column vectors in A
\[ \begin{bmatrix} 2 \\ 3 \\ 5 \end{bmatrix} x1 \,+\, \begin{bmatrix} 1 \\ 1 \\ 7 \end{bmatrix} x2 \,+\, \begin{bmatrix} 3 \\ 4 \\ 12 \end{bmatrix} x3 \]
Basis and Rank
The independent columns using which the other columns can be expressed as a combination of is known Basis for column space. The number of independent columns is known as Rank
\[ \begin{bmatrix} 2 & 1 & 3 \\ 3 & 1 & 4 \\ 5 & 7 & 12 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 3 & 1 \\ 5 & 7 \end{bmatrix} \begin{bmatrix} 0 & 0 & 1 \\ \rowcolor{pink} 0 & 1 & 1 \\ \end{bmatrix} \]