Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 4th Quiz April 7, 2005 Problem 1 Find an orthogonal basis for the plane x - y + z = 0 , viewed as a subspace of R 3 . 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 k ~u - ~w k ≤ k ~v - ~w k
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Unformatted text preview: ~v ∈ V. b. In the inequality above, is such a ~u unique for every ~w ? If so, is the assign-ment ~w 7→ ~u linear? If it is, find the matrix A of this linear transformation in the standard basis of R 3 . Problem 3 Find the Q-R factorization of the matrix 4 6 10 0 8 12 0 0 14 . 1...
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