An exam with m questions is given to n students. The instructor collects all the grades in a n x m matrix G, with G_ij the grade obtained by student i on question j. We would like to assign a difficulty score to each question, based on the available data.

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How would you compute a rank-one approximation to G? State precisely your answer in terms of the SVD of G.

Assume that the grade matrix G is well approximated by a rank- one matrix sqT, with s & R" and q = Rm (you may assume that both s, q have non-negative components). Explain how to use the approximation to assign a difficulty level to each question. What is the interpretation of vector s?