Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 4th Quiz April 8, 2005 Problem 1 Find an orthogonal basis for the plane x - y + z = 0 , viewed as a subspace of R 3 . Solution So it is easy to see that the basis for the plane is v 1 = 1 1 0 , v 2 = 1 0 - 1 Now let’s use Gram-Schmidt algorithm to get the orthogonal basis of it. v 1 = 1 1 0 , then | v 1 | = v 1 · v 1 = 2 So we have u 1 = v 1 | v 1 | = 1 2 1 1 0 And v 2 = v 2 - Proj u 1 v 2 = v 2 - ( v 2 · u 1 ) u 1 = v 2 - 1 2 v 1 = 1 2 - 1 2 - 1 , and | v 2 | = 3 2 Therefore, u 2 = v 2 | v 2 | = 2 3 1 2 - 1 2 - 1 So the orthogonal basis for the plan is 1 2 1 1 0 , 2 3 1 2 - 1 2 - 1 1
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Problem 2 Let e 1 , e 2 , e 3 be the standard basis of R 3 . Consider the plane V spanned by e 1 and e 2 . a. For a given vector w = ( a, b, c ) R 3 , calculate the vector u V that minimizes the distance between V and w , i.e. find u V such that u - w v - w v V. Solution u = Proj V w = a b 0 b. In the inequality above, is such a
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