Another way to think the matrix mulitply

Usually, we think the matrxi multiply as follows: \[ Ax = b = (A_{1,:}b_1,A_{2,:}b_2,...,A_{n,:}b_n)^T \] Also, we can get the result by : \[ Ax = b = \Sigma_i^n (x_iA_{:,i}) \]

This kind of operation is just like the linear combination. So when consider the solutions of such \(Ax = b\), it is equivalent to find whether we can find any linear combination to construct the \(b\).

Take an example: \[ A = \left\{ \begin{matrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ a_{3,1} & a_{3,2} \end{matrix} \right\} \qquad x = \left\{ \begin{matrix} x_{1,1} \\ x_{2,1} \end{matrix}\right\} \qquad b = \left\{ \begin{matrix} b_{1,1} \\ b_{2,1} \\ b_{3,1} \end{matrix}\right\}\\ Ax = b \] The equation has a solution if and only if \(b\) lies on the plane where \(A_{:,1} and A_{:,2}\)linear combination