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.