deep learning book notes

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