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# midterm - MATH 110 MIDTERM LECTURE 1 SUMMER 2009 Name 1(10...

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Unformatted text preview: MATH 110 - MIDTERM LECTURE 1, SUMMER 2009 July 16, 2009 Name : 1. (10 points) Give two equivalent definitions (or characterizations) of each of the following: (a) A list ( v 1 ,...,v n ) is linearly dependent (b) An operator T ∈ L ( V ) can be represented by an upper-triangular matrix (c) A scalar λ ∈ F is an eigenvalue of an operator T ∈ L ( V ) (d) A vector space V is the direct sum of subspaces U and W 2. (10 points) Determine whether the following statements are true or false. You do not need to provide justifications for your answers. There will be 1 point subtracted for each incorrect response, so do not guess just for the sake of guessing. (a) Any operator on a real vector space has a 1-dimensional invariant subspace. (b) If V = U ⊕ W and p ( z ) = z 2 , then p ( P U,W ) = P U,W . (c) If v 1 ,...,v n ∈ V , then span( v 1 ,...,v n ) is n-dimensional. (d) Any operator on a 10-dimensional complex vector space has a 4-dimensional invariant subspace. 3. (15 points) Let n be a positive integer and let...
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midterm - MATH 110 MIDTERM LECTURE 1 SUMMER 2009 Name 1(10...

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