If you don’t want to sit through this whole thing, the video is really split up into two parts. The first one ends around minute 20. And at the end of the day the most important part of Part two are the two equations listed at the end (the video is really just the context and explanations that make those equations make sense!)
Jan 23
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Hey guys! This is Taylor commenting:
I would also like to point out that the “Laws” that I mentioned in the run up to the theorem about the number of equations versus number of unknowns are all true, but they are examples that the book wanted you to prove: a more complete set of laws, that is, I think a little more helpful is:
If Rank(A)=n, then the system is consistent.
If rank(A)< m, then system has none or infinite solutions
If rank(A)=m, then system has no solutions or exactly one.
* I would also like to amend a quick statement that I made: I mention the fact that if Rank(A)<n, there is also the possibility that the solutions are infinite: this is true because one equation on the bottom in the rref could be for example: 0+0+0=0, but this would only make it infinite if it meant that there were then free variables that could not be solved for in the other equations: I did not specify that in the video.