exsheet3 - Mathematical Methods I Dr Gordon Ogilvie Example...

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Mathematical Methods I Natural Sciences Tripos, Part IB Dr Gordon Ogilvie Michaelmas Term 2008 Example Sheet 3 1. Use the Cauchy–Schwarz inequality and the properties of the inner product to prove the triangle inequality | x + y | l | x | + | y | for a complex vector space, where | x | is the norm of the vector x . Under what conditions does equality hold? 2. Given a set of vectors u 1 , u 2 , . . . , u m ( m g n ) that span an n -dimensional vector space, show that an orthogonal basis may be constructed by the Gram–Schmidt procedure e 1 = u 1 , e r = u r r - 1 s s =1 e s · u r e s · e s e s for r > 1 . What is the interpretation if any of the vectors e r vanishes? Find an orthonormal basis for the subspace of a four-dimensional Euclidean space spanned by the three vectors with components (1 , 1 , 0 , 0), (0 , 1 , 2 , 0) and (0 , 0 , 3 , 4). 3. What does it mean to say that the vectors
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exsheet3 - Mathematical Methods I Dr Gordon Ogilvie Example...

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