(I am trying to understand a theorem from my textbook)
[Theorem]: Consider the inhomogeneous system with the rank of $k,\ (k<n)$:
$$\begin{matrix}a_{11}x_1+a_{12}x_2+…+a_{1n}x_n=b_1\\a_{21}x_1+a_{22}x_2+…+a_{2n}x_n=b_2\\……\\a_{n1}x_1+a_{n2}x_2+…+a_{nn}x_n=b_n\end{matrix}$$
The homogeneous adjoint of this system is then:
$$\begin{matrix}\bar{a}_{11}y_1+\bar{a}_{21}y_2+…+\bar{a}_{n1}y_n=0\\\bar{a}_{12}y_1+\bar{a}_{22}y_2+…+\bar{a}_{n2}y_n=0\\……\\\bar{a}_{1n}y_1+\bar{a}_{2n}y_2+…+\bar{a}_{nn}y_n=0\end{matrix}$$
We conclude that:
A necessary and sufficient condition for the inhomogeneous system to have a solution is that the vector $B=(b_1,b_2,…,b_n)$ be orthogonal to all the vectors obtained as solutions of the homogeneous adjoint system.
[Question]:
I have read the derivation of this theorem and understand all the calculations to obtain it, but I can't understand it intuitively. Why is the orthogonality here determines the existence of the solution? Is there any narrative or geometry based explanation for this ?
Best Answer
To find the solutions of $Ax=b$, first we denote $A$ as a set of column vectors $(\alpha_1,\alpha_2,\cdots,\alpha_n)$, so a necessary and sufficient condition for the inhomogeneous system to have a solution is that $\exists x$, s.t.
$x_1\alpha_1+x_2\alpha_2+\cdots+x_n\alpha_n=b$, which means that $b\in L(\alpha_i)$.
Also we know that $V=U\oplus U^{\perp}$, so $b\in L(\alpha_i)\Longleftrightarrow b \perp v,\forall v \in L(a_i)^{\perp}$.
Actually you can find a set of orthogonal basis of {$\alpha_i$}, so to have the solution of inhomogeneous system $Ax=0$
we can expand {$\alpha_i$} into the basis of $V$, and we can get the basis of the solution.
Plus we have $\ker \phi^{*}=($ $\mathrm{Im}$ $\phi$)$^{\perp}$.