Math 110 Professor K A Ribet Midterm Exam September 26 2002 Please put away all books calculators digital toys cell phones pagers PDAs and other electronic devices You may refer to a single 2 sided sheet of notes Please write your name on each sheet of paper that you turn in Don t trust staples to keep your papers together The symbol R denotes the field of real numbers In this exam 0 was used to denote the vector space 0 consisting of the single element 0 These solutions were written by Ken Ribet Sorry if they re a little terse If you have a question about the grading of your problem see Ken Ribet for problems 1 2 and Tom Coates for 3 4 1 9 points Let T R4 R3 be the linear transformation whose matrix with respect to 1 1 7 5 the standard bases is A 1 0 5 3 In the book s notation T LA 0 1 2 2 Find bases for i the null space and ii the range of T The null space consists of quadruples x y z w satisfying three equations of which the first two are x y 7z 5w 0 and x 5z 3w 0 It turns out that the third equation is the sum of the first two so we can forget it If you don t notice this circumstance you ll still get the right answer Replace the first equation by the sum of the first two leaving the second alone We get the two equations y 2z 2w 0 x 5z 3w 0 The interpretation is that z and w can be chosen freely and then x and y are determined by z and w If we take z 1 w 0 we get the solution 5 2 1 0 With the reverse choice we get the solution 3 2 0 1 These form a basis for the null space Once we know that the null space has dimension 2 we deduce that the range has dimension 2 as well In fact it consists of the space of triples a b c with c a b A basis would be the set containing 1 0 1 and 0 1 1 Of course there are other correct answers bases aren t unique Let me stress that R T lies in 3 space If your answer consists of vectors in R4 you ve messed up Note from Ribet An answer that just has a bunch of numbers with no explanation as to what is going on is very unlikely to receive full credit You need to tell the reader me in this case what you are doing 2 9 points Let V be a vector space over a field F Suppose that v1 vn are elements of V and that w1 wn wn 1 lie in the span of v1 vn Show that the set w1 wn 1 is linearly dependent Let W be the span of v1 vn Then W is generated by n elements so its dimension d is at most n for example by Theorem 1 9 on page 42 If the vectors wi were linearly independent the set w1 wn 1 could be extended to a basis of W This is impossible because all bases of W have d elements 3 10 points Let W1 and W2 be subspaces of a finite dimensional F vector space V Recall that W1 W2 denotes the set of pairs w1 w2 with w1 W1 w2 W2 This product comes equipped with a natural addition and scalar multiplication w1 w2 w10 w20 w1 w10 w2 w20 a w1 w2 aw1 aw2 This addition and scalar multiplication make W1 W2 into an F vector space There was no requirement or expectation that students verify this point 1 Check that the map T W1 W2 V w1 w2 7 w1 w2 is a linear transformation This is fairly routine For example T a w1 w2 T aw1 aw2 aw1 aw2 a w1 w2 aT w1 w2 A similar computation shows that T of a sum is the sum of the T s 2 Prove that N T 0 if and only if W1 W2 0 The space N T consists of pairs w1 w2 with w1 w2 0 Hence w2 is completely determined by w1 as its negative The wrinkle is that w1 has to be in W1 while w2 w1 has to be in W2 Thus w1 has to be in both W1 and W2 i e in the intersection of the two spaces The null space N T is in 1 1 correspondence with W1 W2 with w1 w2 N T corresponding to w1 W1 W2 and w W1 W2 corresponding to w w N T In particular N T 0 if and only if W1 W2 0 3 Show that dim W1 W2 dim W1 dim W2 dim V Consider T W1 W2 V We have dim W1 W2 dim N T dim R T dim W1 W2 dim R T Here I have used the fact that the identification between N T and W1 W2 that we discussed above is a linear identification one that respects addition and scalar multiplication Hence it preserves dimension For this problem we have to use another fact namely that dim W1 W2 is the sum of the dimensions of W1 and W2 This follows from the Ribet s 110 first midterm page 2 fact that we get a basis for the product by taking the union of 1 0 and 0 2 where the i are bases of the Wi i 1 2 This fact gives dim W1 W2 dim W1 dim W2 dim W1 W2 dim R T dim W1 W2 dim V The desired inequality follows 4 12 points Label the following statements as being true or false For each statement explain your answer 1 The span of the empty set is the empty vector space There is no empty vector space The span of the empty set is 0 2 If v is a vector in a vector space V that has more than two elements then V is spanned by the set S w V w 6 v The span of S contains S so it contains all w different from v Does it contain v as well Yes indeed choose w V different from 0 and v this choice is possible because V has more than two elements Write v v w w The vectors v w and w are both in S neither is v Hence v is in the span of S Our conclusion is that the span of S contains all of V so the assertion is true 3 Suppose that T V W is a linear transformation between finite dimensional Rvector spaces If dim V dim W and w lies in the range of T then there are infinitely many v V such that T v w If w lies in the range of T then there is some v V such that T v w Fix this v and notice that T v 0 w if and only if …