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set_3_problem

# set_3_problem - 1 2 2 1 1 1 and 1 1 Problem 3 Prove that...

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Problem Set # 3 MMAE: 501 Engineering Analysis I Kevin W. Cassel Mechanical, Materials and Aerospace Engineering Department Illinois Institute of Technology 10 West 32nd Street Chicago, IL 60616 [email protected] Problem Reference Problem # 1: Hilderbrand, Chapter 1, Problem 53 Problem # 2: Hilderbrand, Chapter 1, Problem 54 Problem # 3: Hilderbrand, Chapter 1, Problem 55 Problem # 4: Jeffrey, Section 4.1, Problem 19 Problem # 5: Jeffrey, Section 4.2, Problem 16 Problem # 6: Jeffrey, Section 4.4, Problem 14 c 2007 Kevin W. Cassel 1

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Problem # 1 Suppose that the system x 1 - αx 2 = c 1 - αx 1 + x 2 - αx 3 = c 2 - αx 2 + x 3 = c 3 is solved iteratively by use of the relations x ( r +1) 1 = αx ( r ) 2 + c 1 x ( r +1) 2 = α ( x ( r ) 1 + x ( r ) 3 ) + c 2 x ( r +1) 3 = αx ( r ) 2 + c 3 . Show that convergence is guaranteed if and only if | α | < 1 2 . Problem # 2 Construct a set of three mutually orthogonal unit vectors which are linear com- binations of the vectors
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Unformatted text preview: { 1 , , 2 , 2 } , { 1 , 1 , , 1 } , and { 1 , 1 , , } . Problem # 3 Prove that the vector v = { 2 , 1 , 2 , } is in the space generated by the three vectors deﬁned in Problem 2 and express v as a linear combination of the se-lected vectors e 1 , e 2 , and e 3 . Problem # 4 Find the eigenvalues and eigenvectors of the matrix 2-1 1 2-1 3 . Problem # 5 Use the Gram–Schmidt orthogonalization process with the given set of vectors to ﬁnd an orthonormal set of basis vectors. -1 2 , 1 1-1 , 1-1 1 . 2 Problem # 6 Reduce the quadratic form to its standard form, and use the reduction to classify it: 3 2 x 2 1-x 1 x 3 + x 2 2 + 3 2 x 2 3 . 3...
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