Paul's Test 1


 * Note: I consider to not consider this a test since, on hindsight, I realized that we never went over the second half material, so it was unfair to give a 24 hour test on new material.  So the below is just a review summary, not a "test of knowledge."

GRADE:

First half: 47/50 Second half: 10/50 Total: 57/100

Comments:

Good for your first test (most college tests average at around 50%). The linear algebra is ok and you owned the first half of the test. But we need to get continuity-Epsilon/delta proofs solid. Physics focuses on real world approximations and the reason why an approximation is valid is if it has arbitrary closeness (specifically epsilon delta closeness). For example, the infamous "small angle" approximation uses arbitrary closeness in its justification using a Taylor's series. Epsilon-delta closeness is the heart of analysis. For this week, I am going to postpone the new homework and we are going to go back to continuity of both multivariable and single variable. I will give you an updated homework assignment this evening.

Once again, really good job on the first half: when we do the BIg test, I know you will get the continuity questions.

1. Good manipulation of terms/application of number theory. 10/10. 2a. Good understanding of intersections and unions. Correct. 5/5. 2b. Correct reasoning and correct choice BUT no counterexample given. 3/5. 3. Good use of complement laws. 10/10. 4. Good use of matrix properties. 10/10. 5. Correctly found an A, B and C and proved that A,B satisfied properties. C, however, is "incorrectly" proven to have no inverse. In the formula for a 2D inverse, you are correct that encoded in the formula is the statement "determinant is 0" which implies C has no inverse. But you did not cite this theorem. A more valid approach would have been to show either C maps to distinct points to the same point (failing injectivity) or C fails to map to a point in R^2 (failing surjectivity). I will take away 1 point (Thanks for finding an A such that A^2=A with A not 0 or identity. Whoops). 9/10. 6. Incorrect proof. Just because something in true for any finite case, it is not necessarily true for the infintite case (this is the basis of analysis, which is the center of physics. It is also where we separate the engineers from the physicts/mathematicians). For example, an intersection of finitely many open sets is still open (ex. (-infinity, 1/2), (-infinity, 1/3),...(-infinity, 1/100) are all open and when all intersected, yield (-infinity, 1/100). But the infinite intersection of (-infinity, 1/n) is the set (-infinity, 0], which is not open.

The correct proof: Let x be a point in the infinite union U. Then x must lie, by definition of union, in one of the U_i for some i. But U_i is open. So we have an open ball around x IN U_i. But! U_i is contained in infinite union U. So the open ball around x in U_i is ALSO a ball in U. Since x was arbitrary (general case), we can always find a ball around x contained in U. since this is the definition of open, U is open.

Please redo this proof in your own words since it is fundamental. You also forgot to answer 6b. 0/10.

7. Good, but you could have also argued straight from definition of closed into of using complement rules (which is sometimes easier). Specifically, a converging sequence of points by definition converges to a point in R^n. So duh. Also, that is correct the reasoning for the empty set, but I prefer an arguement using the words "vacuously true." Example: All girls in the math department at Stanford are pretty. Specifically, this means for any girl I find who is in the math department, I can say that girl is pretty. But THERE are no girls here! We can also say, at the same time,all girls in the math department at Stanford are ugly. 10/10

8. Last step untrue. Not sure what you are doing here. 9. Sorry. No. 10. Dude, continuity proof! It is the same proof as continuous in components. Ah well, at least we know what we should cover.