Markers Comments

 Some students had
difficulty determining the field for the scalar when showing linearity
(for instance in question 6, limiting the scalar to the reals instead of
to whatever field applies or in question 1 again using reals instead of
complex).  Also some students had a bit of difficulty with question 4c)
part ii when finding T[fcn] w.r.t. gamma.  But overall the assignment
was done fairly well.

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1) There is a significiant number of students who don't understand the
concept of onto very well - the answers to 1(b) were all over the
map. There were, however, some good answers - people constructed a
basis through the linear transformation, that sort of thing. There
were also a few people who did the right thing but didn't present it
well. Nevertheless, I got the sense that some people weren't clear on
on to at all.

2) Very, very few people did 1(c) correctly - a lot of people showed
that anything in N(L) is in span({v}) perp, but not many people showed
they were actually equal. Lots of hand waving, not very much rigour -
and the simple solution that is in the answer set was in maybe 3 or 4
of the 31 papers I have this week.

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1a) Many failed to say by Properties of Inner Product.  I did not deduct a 
mark but I made a note of it on their assignments.



1b) was poorly done overall.  Quite a few people do not have a good 
understanding of what "onto" means.

Solutions should start like: let c be a complex number, then find a w part of 
V such that Lv(w)=c.

This way it is shown that for any c being a complex number, we can find a 
vector w from V such that Lv(w)=c thus making the transformation onto.

Many ppl said that Lv(w)=(w,v)=w1v1+...+wnvn.  They then concluded that any c 
complex number can be found.  In my opinion this does not show that ANY 
complex number has a corresponding w such that Lv(w)=c.

A few others simply stated that "any complex number" can be expressed as an 
inner product, so it is onto. In my opinion this is also missing a lot.



6) a few ppl had some small difficulties.  I received a few answers which were 
of the sort:  T(u+v)=T|u(u+v)=T|u(u)+T|u(v)  and T(cu)=T|u(cu)=cT|u(u)  thus 
T|u is linear.

You cannot state that T|u(u+v)=T|u(u)+T|u(v) OR T|u(cu)=cT|u(u) since this is 
in fact what you need to prove!  People were writing this down and concluding 
(wrongly) that T|u is therefore linear!

As well, a few people failed to give justification as to why T(u+v)=T(u)+T(v)  
If they did not mention that this is because T is linear I deducted one mark.


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    In question one, some students made unwarranted assumptions about the
question.  Many assumed, for example, that the inner product in the question
was the standard inner product.  In doing so, many also implied that V is
the one-dimensional complex field.

    In questions three and four, some students solved the incorrect matrix,
often making the question more complicated.  The vast majority of the
students were referring to the matrix representation of the matrix as A.
Those who wrote the matrix with the basis in the superscript and subscript
generally avoided these errors.

    The question with the most problems was 4cii.  Many students failed to
convert the polynomial to a coordinate of the appropriate basis, instead
using the standard basis.  Again, this generally only occurred with students
who represented the matrix with the letter A, rather than a more descriptive
name.

Marking Scheme

Assignment 3         total 45 marks

Problem 1:
          5 marks for each of a,b,c                              total 15 marks

Problem 3:
          5 marks each for b part i and c part ii                total 10 marks
Problem 4:
          5 marks each for b part i and c part ii                total 10 marks

Problem 5.  part a i    5 marks                                  total 5 marks

Problem 6.  part a     5 marks                                   total 5 marks