Math 110 Midterm Exam Professor K A Ribet September 29 2003 Please put away all books calculators electronic games cell phones pagers mp3 players 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 Explain your answers in full English sentences as is customary and appropriate Your paper is your ambassador when it is graded All problems were worth 10 points 1 Suppose that F is the field of rational numbers Let V P100 F be the F vector space consisting of polynomials over F of degree 100 Let T n n X X d i iai xi 1 Find the ai x 7 V V be the differentiation operator dx i 1 i 0 nullity and the rank of T The kernel null space of T is the 1 dimensional vector space consisting of constant polynomials Hence n T 1 and r T dim V n T 101 1 100 Suppose now instead that F is the field Z5 consisting of the integers 0 1 2 3 and 4 mod 5 What are the nullity and the rank in this case The null space includes some non constant polynomials x5 is a good example Note that 5x4 0x4 0 On reflection you see that f x has derivative equal to 0 if and only if f contains only monomials xi where i is divisible by 5 Hence N T is generated by 1 x5 x10 x100 it has dimension 21 Thus r T 101 21 80 in this situation 2 Let V and W be vector spaces over F with V finite dimensional Let X be a subspace of V Establish the surjectivity onto ness of the natural map L V W L X W that takes a linear transformation T V W to its restriction to X One has to show that every linear map X W can be extended to V For this it is convenient to choose a basis of V by beginning with a basis x1 xd of V and then completing it to a basis x1 xd vd 1 vn of V If T X W is a linear transformation one can define U V W by the formula X X X U ai xi bj vj T ai xi This sense because each vector of V may be written uniquely in the form X makesX ai xi bj vj It is routine to check that U is a linear map and that U coincides with T on X 3 Let V be a finite dimensional vector space over F Let V be the vector space dual to V Let T V F be a linear map Show that there is a vector x V such that T f f x for all f V This problem amounts to the surjectivity of the natural map V V that is discussed in 2 6 of the book In the notation of that section we are trying to show that there is an x V so that T x I am crossing my fingers and hoping that people will explain something of what is going on and not say simply that the wanted result was proven in class or say that it follows from Theorem 2 26 of the book Take a basis v1 vn for V and let fi be the vectors in the basis dual to v1 vn Let ai t fi for i 1 n and set x a1 v1 an vn A quick computation should show that this x has the desired propoerty H110 first midterm page 2
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