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| {{attachment:column-picture-solution.png}} | {{attachment:column-picture-solution.png||width=200}} == Matrix Picture == {{{ ┌ ┐ ┌ ┐ ┌ ┐ │ 2 -1│ │ x│ │ 0│ │-1 2│ + │ y│ = │ 3│ └ ┘ └ ┘ └ ┘ }}} |
2 Equations, 2 Unknowns
Introduction
Consider the below system of equations:
2x - y = 0 -x + 2y = 3
This can be envisioned in three ways.
Row Picture
The equations can be plotted together, and all intersections of the lines are solutions to the system.
Column Picture
The system is equivalent to the following linear combination of columns.
┌ ┐ ┌ ┐ ┌ ┐ │ 2│ │-1│ │ 0│ x│-1│ + y│ 2│ = │ 3│ └ ┘ └ ┘ └ ┘
Column 1 is represented as a vector to (2,-1); column 2 as (-1,2).
Any combination of these vectors that leads to (0,3) is a solution to the system. For this system, that solution is 1 of column 1 and 2 of column 2.
Matrix Picture
┌ ┐ ┌ ┐ ┌ ┐ │ 2 -1│ │ x│ │ 0│ │-1 2│ + │ y│ = │ 3│ └ ┘ └ ┘ └ ┘