MATH 21B Midterm: MATH 21B Harvard 21b Spring 15Solution8
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Instructions: please print your name at the top of each page of this exam, this exam consists of 7 questions. You should answer all parts of all questions: write your answers on the exam paper. / 100 (1) (a) the trace tr(a) of a square matrix a is the sum of the diagonal entries of a. Show that if a and b are n n matrices then tr(ab) = tr(ba) Deduce that similar matrices have the same trace. (5 points) (b) show that (xi xj) = yi>j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (hint: to show that the determinant is divisible by xi xj, show that it vanishes when xi xj = 0. ) (5 points) 2 n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ((question 1 continues on the next page)) (c) de ne inner products on the vector spaces p2(r) and p1(r) by setting. Find the rank r of t , the non-zero singular values 1 .
