202W.2C What do pivots mean? 3x3 examples

Pivots give an essential measure of "independence" in a linear system. Here's what that can look like in several 3x3 cases.

[darshan]

I'm late to comment but I'm here to say, you're on 3B1B level minus the animations (but you still have visualisations!) and abstraction. Great work.

[davidkwon]

In 3X3 system, Rows view point :
1. Three pivots = three hyperplanes intersect in one point (the system is consistent, having a unique solution)
2. Two pivots = only two hyperplanes matter (third one is redundant )
if the third row is 0=0, the third row does not screw up the system [immersivemath.com ch5. Figure 5.12 second], if I remove one of the equations, the solution set would not change at all. Infinite (a line's worth of) solutions.
if the third row is 0=7, the third row ruins us. Inconsistent. no solution. [immersivemath.com ch5. Figure 5.13 first]

[davidkwon]

#Pivots
1. Columns = Variables (pivots ~~ basic variables: determined by the system not by universe) : To what extent does each variable matter in solution set?
2. Rows = Equations = Hyperplanes ~~ How many equations tell their own independent story?

[davidkwon]

#Pivots: #Basic (bound, determined) variables, OR #Independent (not redundent) hyperplanes