MISTAKES: Q23 should be False. For example v1 is the zero vector and v2, v3 are any two orthogonal non zero vectors. Good luck studying for you Linear Algebra Final! Check out my Linear Algebra playlist for more practice problems! ru-vid.com/group/PLscpLh9rN1Rfo0ifw9RZFoJ2Te2jk_pwX
Q28. Does a diagonal matrixv with non-zero components scale. You said true but it could be the identity matrix in which case B is left unchanged/unscaled.
@@DrWeselcouch Last night's discussion with my teenage son on this went like this: Me: If you scale a number by 1 so that it doesn't change, are you actually scaling it? Son: Yeah. Me: So if you rotate something by zero degrees, then that's a rotation? Son: If you rotate something by 360 degrees, is that a rotation? Me: I concede the argument.
This is going to be insanely helpful! I will recommend my former students to watch this since I don't teach linear algebra myself and haven't done it for over a decade. I will also watch it to review it for myself. Thank you!
@@DrWeselcouch Yes, and that matrix isn't in echelon form. Hence the rows of A are a basis for Row A if A is in echelon form, but not only if A is in echelon form.
Just to be clear, the statement by itself is false; the counterexample you gave shows that clearly. I was just questioning whether "This is only true if A is in echelon form" holds.
Nice video. I'd prefer if there was a slightly longer pause between asking the questions and showing the answer, even a second longer, so I have time to pause and consider it before you reveal the solution.
Thanks for the feedback! I'll keep that in mind for my next T/F video. There is a PDF with all the questions in the description if you want to try the questions before watching the video.
Thank you professor. I have a wonder about the question 12. You say that this would be true if A had a pivot position in every column. Do you mean every column as well as row? Because for example A = [1, 0; 0,1; 0,0] has a pivot in every column but b = (0,0,1) (a vector in R^3) has no solution, yes? Or is the question saying, if there is a solution, it must be unique?
I have a question on 82nd question, on 56:09, so what if our matrix is an orthogonal matrix, then A^T would be equal to A^-1, and the eigenvalues of A^-1 is 1/ λ. So wouldn't our eigenvalues change in this case ?
Q23 is wrong, right? v1 could be 0, v2 wouldn't be a multiple of v1, you can take v3 linearly independent of v2, and {v1, v2, v3} wouldn't be linearly independent?
@@DrWeselcouch sorry, my textbook defines orthogonal set as a set of non zero vectors such that they’re pairwise orthogonal. That’s why I am confused. Thanks for the reply 👍