For problem A, one way to characterize a point X is inside the triangle is that: If AX meets BC at P, BX meets CA at Q and CX meets AB at R, then all signed ratios BP/PC, CQ/QA and AR/RB are positive.
For problem D, that "P divides AB in the ratio 2:1" means AP:PB=2:1.
I thought I will not be grading problem 4 but I saw common mistakes in it so I graded everything (hence the delay). They did generally a good job in the first three problems; the only common mistakes were:
1. When finding the inverse of a transformation in problem 2, they apparently plugged in everything to a formula in the book not being careful what each term is supposed to be! If $y= t(x) = Tx + b$ to find the inverse one should "isolate" x so $x = T^{-1} y - T^{-1} b$, whereas often people had calculated $T^{-1} x + T^{-1} b$ instead. By the way, a formula in the book with some U and some a is not a universal convention obviously, so if you want to use the same exact naming you should define what $U$ and $a$ are (which will avoid such mistakes as well).
2. When finding the reflection with respect to the line, problem 3(c) $\theta$ is a clockwise rotation so $sin (\theta)$ is negative. In problem 4 part (d): as I see by "transformation" the book does not only mean the affine symmetries of $R^3$ and the term is just a substitute for "function" (I guess this choice is made to emphasize the fact that we are using a group of "transformations" to determine the geometry of the space we are studying... after Klein!)
Grade = max(G1, G2);
G1 = 10% (HW) + 20% (MT1) + 20% (MT2) + 50% (Final);
G2 = 15% (HW) + 10% (worse MT) + 25% (better MT) + 50% (Final).
First of all, I graded problems A, B and D only, since C was a repetition of what they were already doing in other problems and I didn't want to put too much penalty for same mistakes.
In problems A and B: Almost all of them know how to find eigenvalues and eigenvectors of a finite dimensional operator, however they made a very common mistake like this:
1 - \frac{5 + \sqrt{10}}{2} = \frac{-3 + \sqrt{10}}{2}i.e. they are not careful in distributing the minus sign over all term and this makes all their computations just wrong! At first I felt guilty for deducing a considerable portion of the grade of these problems for computational mistakes but then I saw that they all have the same mistake above, they should really try to avoid. Also there were a few papers in which calculators were used. When doing linear algebra, one should be very careful about using calculators since the computation error grows higher by the number of operations (that's what people study in the field of numerical analysis)! To us \sqrt{17} is a number we can happily leave on its own. As a matter of fact, I did not accept 0.98 as 1 since they are not identical!
Problem D: Their work was overall impressive. A few mistakes: 1. One should watch the order one gives to the eigenvalues and keep that order on corresponding eigenvectors and when forming the matrix P. 2. When graphing the hyperbola, the branches are supposed to intersect the major axis not the minor axis!!
I guess if they try to be more careful in algebraic manipulations they are doing good. I had even more sheets than last time with no names on them. Obviously, if they don't write their names I cannot enter their grades into my list.
Problem 1. No one has cared about the equations they were given. Many didn't even realize the equation is that of an "infinite" cylinder and wrote that there are intersections such as rectangles, etc. Worst is that when they were thinking of a bounded cylindrical shape, many have confused the "part of an ellipse" with parabola. At any rate, whoever has written down words like "line, circle" has got at least 1/5 (this is also what they have got for instance when they have talked about the parabola case, but have not written anything about the situation of the plane). Many have not discussed the situation of the plane at all, however I only deducted 1 point from people who have written correct answers and had discussed the situation of the plane with respect to the cylinder in "English" but had no analysis on the parameters c and d.
Problem 3: I deducted one point from many; this was the case when they had solved for t_1 t_2 but had completely neglected the case t_1 t_2 = 0. Likewise, for the second part of this problem.
Problem 4 and 5: There wasn't any common mistake.
Can you please ask them to staple their papers, I will deduct a point from people who forget this from next time. And finally there was one work with no names on it. I have put it on top of others.