Why equations are called independent and dependent?

Question

I am not sure if I am trying to know everything to detail, but why for example two equations are called independent if they have only single solution? On what they think about when they say single solution, final values of equation variables? How do the equations depend on each other if they have more than single solution in common? Why we used exactly that terms dependent and independent? Is math full of facts that if we want to understand it, we just need to accept those things exactly as the they were written by math scientists or there is some logic behind all of that?

Answer 1

Why for example two equations are called independent if they have only single solution?

Assume you have two equations $$ a_i^\top x = b_i \\ a_j^\top x = b_j $$ for $i \ne j$ of the system $A x = b$.

The equations are linear dependent, if the row vectors $a_i^\top$ and $a_j^\top$ satisfy the equation $$ c_i a_i + c_j a_j = 0 $$ for $(c_i, c_j) \ne 0$. Assuming $c_i \ne 0$ then $$ a_i = (-c_j / c_i) a_j $$ so $a_i$ is a scalar multiple of $a_j$.

The vectors $a_i$ and $a_j$ can be interpreted as normal vectors of a hyperplane with distance $d_i = (1/\lVert a_i \rVert)\, \lvert b_i \rvert$ to the origin.

So linear dependency means those hyperplanes are parallel. They are either the same plane (having same distance from the origin) or they do not intersect (having different distance from the origin).

Note that a solution $x$ is element of the intersection of all hyperplanes of the system.

Both cases result in either loosing one equation, resulting in an infinite number of solutions, because at least one unknown is free or having no solution at all.

So if there is a single solution, both equations have to be linear independent.