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.
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