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midterm2004_1 - 2.035 Midterm Exam Part 1(In class Spring...

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2.035: Midterm Exam - Part 1 (In class) Spring 2004 1.5 hours You may use the notes you took in class but no other sources. Please give reasons justifying each (nontrivial) step in your calculations. ——————————————————————————————————————————— Problem 1: Let R be a 3-dimensional Euclidean vector space and let { f 1 , f 2 , f 3 } be an arbitrary (not necessarily orthonormal) basis for R . Define a set of vectors { e 1 , e 2 , e 3 } by e 1 = f 1 , e 2 = f 2 + c 21 e 1 , e 3 = f 3 + c 31 e 1 + c 32 e 2 . i) Calculate the values of the scalars c 21 , c 31 and c 32 that makes { e 1 , e 2 , e 3 } a mutually orthog- onal set of vectors. ii) Is the set { e 1 , e 2 , e 3 } linearly independent? iii) Does { e 1 , e 2 , e 3 } form an orthonormal basis for R ? ——————————————————————————————————————————— Problem 2: Let R be the Euclidean vector space consisting of “trigonometric
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