Assignment 2

                        Marking Scheme and TA Comments


Question 1:

Some students didn't show why 
  (|x1| + |x2| + ... + |xn|)^2 >= (|x1|^2 + |x2|^2 + ... + |xn|^2) 
or equivalently why
   (|x1| + |x2| + ... + |xn|) >= sqrt(|x1|^2 + |x2|^2 + ... + |xn|^2) 
I think this is the core part of this proof. So, I took off 2 points if
students simply stated this fact without showing why.


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Question 2:

Some students forgot the absolute value on each component when 
dealing with norms. For example, 
   || X ||_1 = |x1| + |x2| + ... + |xn|
not 
   || X ||_1 = x1 + x2 + ... + xn 
This mistake happened because they only considered the case for which all 
xi are positive.  I take off 1 point.


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Question 3:

I gave them 2 marks for each of valid value of x and y and 1 mark for
the calculation to verify the result.


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Question 4:

I gave them 
  - 3 points for showing that ||D||p = max {|d_i|}
  - 1 point for applying previous result to D^(-1)
  - 1 point for cond_p(D) = ||D||p ||D^(-1)||p
Most students did very well. 

Some
common errors:
  - Not proving that ||D||p = max {|d_i|} (-3 points)
  - Assuming some arbitrary norm (such as the 2-norm) in place of 
    ||D||p (-2 points)
  - only showing that max{|d_i|} is an upper bound for ||D||p
    This is insufficient to prove that the upper bound can be
    reached. For example, 2 is an upper bound of cos(x), but there is 
    no x that makes cos(x) = 2) (-1 point)
  - only showing that ||D||p >= max {|d_i|}
    (by providing a vector x (x=e_i) such that ||Dx||p = max{|d_i|}. 
    This is also insufficient because it only proves that ||D||p is at least
    max{|d_i|}) (-2 points)


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Question 5:

  * most students did well in this question
  * they got 5 if they did everything completely right
  * 4 if they got almost everything
  * 3 if they were on the right track
  * a few students said they would prove something by
    contradiction but did not contradict anything and just ran the
    statement forward


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Question 6:

I gave them 3 marks for a valid example and 2 marks for verifying the
result.  If the norm used in the calculation is not specified or a mix
of different norms is used, I deducted one mark.