mt2_2008_sols - Ribets Math 110 Second Midterm problems and...

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Ribet’s Math 110 Second Midterm, problems and abbreviated solutions Please put away all books, calculators, and other portable electronic devices—anything with an ON/OFF switch. You may refer to a single 2-sided sheet of notes. When you answer questions, write your arguments in complete sentences that explain what you are doing: your paper becomes your only representative after the exam is over. All vector spaces are finite-dimensional over the field of real numbers or the field of complex numbers. 1. Suppose that T is an invertible linear operator on V and that U is a subspace of V that is invariant under T . If v is a vector in V such that Tv U , show that v is an element of U . Quick solution: Let u = Tv . Because the restriction of T to U is invertible, there is a unique v U such that Tv = u . Since Tv = Tv and T is invertible, we have v = v . Hence we have v U . 2. Suppose that T is a linear operator on V and that V is an inner-product space. Let T * be the adjoint of T . Show that 0 is an eigenvalue of T
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