8:00 What if we permuted rows 1 and 3? I supposed we have to change the column space for [0 1 1]. Is that correct? Thank you for all your great videos you made! I look forward to watch them all and I definitely promote your channel on the web.
Okay so regarding column spaces, I can see that by adding a multiple of one row to another, columns are getting sheared. I know that shearing won't change dimensions. So the column space's dimensions would remain the same but the column space might change.
Recalling the first lectures that emphasized the geometric interpretation of "travelling" to villages. Say going from village A to village B. It seems to me that the NULL space are all the non repeating detours that allow me to travel to the same village. In the context of a taxi driver, it would be all the different scenic routes and still arrive at our destination. The later analogy breaks down a little because I would like to pay the same fare all the time and I do not believe the taxi drive would be too happy.
Actually, your intuition corresponds to a very specific and important application of the null space. Check out this lecture by Professor Strang: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-6-wh6yvk6uc.html Also, I really like how you capitalize NULL. After years of C development, when I type "null" I think I'll get a syntax error.
