Linear Algebra 8f: A Tall Linear System

[stgr]

Hi,
Thanks! This is great stuff. At the beginning of the video you said you "made it easy" and the system had a unique solution where [x, y] = [1, 1]. Then, you changed the right side so there was no solution. In explaining this geometrically, you analogized to R3 with 2 vectors and asked, "What are the chances that a 3rd, randomly selected vector will lie in the same plane spanned by the first 2 vectors? Very unlikely." (I paraphrased that a bit.)
So, is it correct to interpret the earlier statement that you 'made it easy' to mean that you CAREFULLY picked a solution (NOT random at all) so that, even though it is unlikely that a RANDOMLY selected vector would lie in the same space spanned by the first two vectors, in this case, it did because of your set up?
And, once you picked that special vector, we are then left with only one solution (in this case [1, 1]) but, if you had carefully picked another vector that lies in the space, perhaps [2, 2, 18, 36] which is twice the first column and twice the second column, then the system would have the unique solution [2, 2].
If this is correct, then there are infinitely many such vectors in the space spanned by the first two and, once we pick the 'solution vector' we lock down [x, y]. Is that correct?
Thanks again for these outstanding videos!
Steve