= 3 Equations, 3 Unknowns =

== Introduction ==

Consider the below system of equations:

{{{
2x - y = 0
-x + 2y - z = -1
-3y + 4z = 4
}}}

This can be envisioned in three ways.


== Row Picture ==

These equations can be plotted as 3D planes. The space of intersection gives all solutions to the system. But this is very difficult to visualize and solve.



== Column Picture ==

{{{
 ┌  ┐    ┌  ┐    ┌  ┐   ┌  ┐
 │ 2│    │-1│    │ 0│   │ 0│
x│-1│ + y│ 2│ + z│-1│ = │-1│
 │ 0│    │-3│    │ 4│   │ 4│
 └  ┘    └  ┘    └  ┘   └  ┘ 
}}}

These columns can be plotted as 3D vectors. Through some combination, the point (0, -1, 4) can be reached. But the solution is trivially x=0, y=0, z=1; note that the target is equivalent to column 3.



----
CategoryRicottone