How to understand the concept behind the equation $\boldsymbol{Ax}=\boldsymbol{b}$ As is know to all, the equation $\boldsymbol{Ax}=\boldsymbol{b}$ can be understand as to find the linear combination coefficient of the column vector of the matrix $A$. At the same time, it can also be explained as to find vector space which is orthogonal to all the row vectors. So I think there must be some connections between this two problems. Maybe we can call them dual problems, Can anyone explain the connections between the two explanations for me?

As far as the equation $Ax=b$ goes, the left-hand-side can the thought of in different ways.
One way is that if $a_1,\ldots,a_n$ are the columns of $A$, and $x=(x_1,\ldots,x_n)$, then
$Ax = a_1x_1+\cdots+a_nx_n,$
which is a linear combination of the columns of $A$. Thus to solve the equation, you need to find the values of $x_1,x_2,\ldots,x_n$ such that this linear combination is equal to $b$.
But the left-hand-side $Ax$ can also be thought of in the following way, which is the usual "row times column" way of thinking about matrix products. If the rows of $A$ are $c_1,\ldots,c_n$, then $Ax$ is the vector whose $i$'th entry is the dot product $c_i\cdot x$. In the special case of the equation $Ax=0$, you get that $c_i\cdot x=0$ for all $i$, and thus that $x$ is orthogonal to all row vectors of $A$.
