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.
--------------------------------------------------------------------
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.
--------------------------------------------------------------------
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.
--------------------------------------------------------------------
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)
--------------------------------------------------------------------
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
--------------------------------------------------------------------
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.