Math 110 Professor K. A. RibetMidterm Exam September 29, 2003Please put away all books, calculators, electronic games, cell phones, pagers,.mp3 players, PDAs, and other electronic devices. You may refer to a single2-sided sheet of notes. Please write your name on each sheet of paper thatyou turn in; don’t trust staples to keep your papers together. Explain youranswers in full English sentences as is customary and appropriate. Yourpaper 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 theF -vector space consisting of polynomials over F of degree ≤ 100. Let T =ddx: V → V be the differentiation operatornXi=0aixi7→nXi=1iaixi−1. Find thenullity and the rank of T .The kernel (= null space) of T is the 1-dimensional vector space consisting ofconstant polynomials. Hence n(T ) = 1 and r(T ) = dim(V ) − n(T ) = 101 − 1 =100.Suppose now instead that F is the field Z5consisting of the integers 0, 1, 2, 3and 4 mod 5. What are the nullity and the rank in this case?The null space includes some non-constant polynomials; x5is a good example.(Note that 5x4= 0x4= 0.) On reflection, you see that f(x) has derivativeequal to 0 if and only if f contains only monomials xiwhere i is divisible by 5.Hence N(T ) is generated by 1, x5, x10, . . . x100; it has dimension 21. Thusr(T ) = 101 − 21 = 80 in this situation.2. Let V and W be vector spaces over F , with V finite-dimensional. LetX be a subspace of V . Establish the surjectivity (“onto-ness”) of the naturalmap L(V, W ) → L(X, W ) that takes a linear transformation T : V → W to itsrestriction 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, . . . , xdof Vand then completing it to a basis x1, . . . , xd; vd+1, . . . , vnof V . If T : X → W isa linear transformation, one can define U : V → W by the formulaU(Xaixi+Xbjvj) := T (Xaixi).This makes sense because each vector of V may be written uniquely in the formXaixi+Xbjvj. It is routine to check that U is a linear map and that Ucoincides with T on X.3. Let V be a finite-dimensional vector space over F . Let V∗be the vectorspace dual to V . Let T : V∗→ F be a linear map. Show that there is a vectorx ∈ V such that T (f ) = f(x) for all f ∈ V∗This problem amounts to the surjectivity of the natural map ψ : V → V∗∗thatis discussed in §2.6 of the book. In the notation of that sec tion, we are trying toshow that there is an x ∈ V so that T = ˆx. I am crossing my fingers and hopingthat people will explain something of what is going on and not say simply thatthe wanted result was proven in class or say that it follows from Theorem 2.26 ofthe book. Take a basis v1, . . . , vnfor V and let fibe the vectors in the basis dualto { v1, . . . , vn}. Let ai= t(fi) for i = 1, . . . , n and set x = a1v1+ · · · + anvn. Aquick computation should show that this x has the desired propoerty.H110 first midterm—page
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