Point distribution:

30 pts for 1-10
20 each for 1-14 a and b
30 for 1-14
The grades ranged from 10 to 95 with a mean of 64.



Some common errors:

A few students think that if the x dot y is zero, then x or y is zero.

Some tried to apply 1-11 but did not seem to understand that x=y only if x dot z
= y dot z for ALL z, not just any z. This happened to a lesser extent in some
applications of 1-10.

In 1-14, many students tried to find x such that y-x is orthogonal to x, not to
any vector in M. They got the right answer by overlooking the possibility that
x=0.

In 1-14 b, many students tried to justify their answer by painting a mental
picture of the vector space rather than by proving it mathematically.

The most widespread error was that MOST students think that if you have xy=xz,
you can just write y=z without considering the possibility of x=0.

Many students who lost points wrote long, rambling "proofs" that did not have
any good reasoning. In general, those who wrote less received the highest
scores.