Just in case anyone needs an explanation (please correct me if I am wrong): For question 3: - "empty" means that the solution set is no solution (i.e., the solution set does not have any solution) - b is a nonzero vector. I think what STEM Support was trying to explain is that because the x vector can be the zero vector, then it cannot be the case that b is a nonzero vector. But because b is a nonzero vector, then the statement must be false. Note: I am still trying to understand and make sense of what STEM Support explained and this is the one which I understand the most.
I don’t think so. If you swapped them, the pivot 15 would not have all zeros below it, which is part of the definition of echelon form. The way it is now has all zeros below the pivots and the pivots move from left to right, even if they skip a column.
Hello. I guess, 3rd question is actually true, because as you mentioned "If we take x as 0, we'd get in ax=b one solution at least, so it wouldn't be empty. If we rewrite this question in a logic form it would be (a->(p or q)). where a is "A is an m x n matrix" and p is "is empty", q is "a span in R^n". Considering this logic statement and that ax=b, has at least one solution, therefore p is false, since it's not empty. But, it is indeed a span in R^n, so q - is true. That leads us that our "or" statement is actually true, because we have a true and false comparison. Therefore, our second part of implication is also true, so 1->1 = true. Correct me if I'm wrong
If A is mxn real matrix, so x must has one of the following solution for Ax=b: (a) unique solution (b) infinite solution, or (c) no solution. Since given statement in true/false question 3 is not always the case, so it is ‘false’. Am I right? Thanks.
1. Should actually be false because a matrix of row echelon form may only have a 1 as the first non-zero entry 3. Unfair question, as b is not specified as containing zeros or real numbers. If homogeneous system, then the span could work
Hey Jesse. Matrices in row echelon form are allowed to have pivots not equal to 1. It's when the matrix is to be in reduced row echelon form that the pivots must all equal 1. As for your comment about question 3, you're absolutely right that it would be true if b was the zero vector, but the instructions for these true false questions (which I cut out of the screenshot, so I should have addressed it in the video) were to mark true only if the statement is always true. In other words, if the statement is ever false, mark false.
STEM Support Huh, yeah I realized later after watching other vids that a lot of people out there use non-zero pivots in REF. My prof was odd I guess and taught it as leading 1s only. Learned something new! Ah and the other one makes sense now
![Midterm 1 True False Easy/Medium/Hard [Passing Linear Algebra]](/img/logo.svg){kind=logo}
