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.