Please clarify: Before, you said the only 3 valid row operations are adding rows together, multiplying a row by a number, and switching rows, NOT COLUMNS. In this video, you say it's okay to switch columns to go from identity to an elementary matrix. Extremely confused, can you please explain? Thanks.
That's a good point. Let me clarify: You cannot perform column operations *for the purposes of determining the null space and solving linear systems*. This is so because column operations alter the relationships among the columns (and between the columns and the RHS) and therefore cannot be used to determine the null space. However, for many other purposes (e.g. QR decomposition), performing column operations is necessary. In conclusion, both row and column operations are of interest, but only row operations can be used to solve systems and determine null spaces.
Thanks for the clarification! One other question - you seem to take for granted that followers of this course have a solid understanding of polynomials and calculus. However, last time I studied graphing equations and derivatives, etc was in high school, a long time ago. I don't want to get sidetracked too far from your linear algebra course, it's exactly what I need for my job (self-taught game programmer building a graphics engine) but could you please direct me to a source (preferably taught by you if there is one) that covers just enough of the basics of polynomials and calculus that will strengthen my grasp of that aspect of L.A., so I can patch up this weak point and get back to your linear algebra course? Please let me know, thanks.
For the last matrix, wouldn't this be possible? 1. Add two of row one to row two 2. Subtract 2/3 of row two from row one 3. Multiply row one with -3 Or isn't it allowed to add/subtract a non-integer number times a row to/from another row, or multiply a row with a negative number? (I never heard you say anything about that) To me, it doesn't make any sense to forbid such operations in an elementary matrix. What is the reason for doing that?
I found this video very confusing at first until I realized that what is being discussed is in-place changes. One way (my way) to think about it is strictly in terms of the original two matrices, and the target. So for example, get 2 of the first column of the left matrix, and 3 of the third column, add them together, and put them in the first column of the target matrix. The approach here is making operations directly on one of the original two matrices (which will be used as the target matrix), and it was when I realized this that I saw why the order of the operations matters.
Correcting myself. After rewatching, I noted that he said, "Looking at this matrix, what are the row operations that yield this matrix if your starting point is the identity matrix." But it is still about making changes in-place, and that is why order matters in some of the examples.
