Chapter 5: Q46E (page 217)
In Exercises 40 through 46, consider vectors in ; we are told that is the entry of matrix A.
46. Find , where V =span . Express your answer as a linear combination of and .
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Among all the unit vectors in, find the one for which the sum of the components is maximal. In the case , explain your answer geometrically, in terms of the unit circle and the level curves of the function.
Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? .
Leg traction.The accompanying figure shows how a leg may be stretched by a pulley line for therapeutic purposes. We denote by the vertical force of the weight. The string of the pulley line has the same tension everywhere. Hence, the forces role="math" localid="1659529616162" and have the same magnitude as . Assume that the magnitude of each force is 10 pounds. Find the angle so that the magnitude of the force exerted on the leg is 16 pounds. Round your answer to the nearest degree. (Adapted from E. Batschelet, Introduction toMathematics for Life Scientists, Springer, 1979.)
Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.
12.
a.Find all n×nmatrices that are both orthogonal and upper triangular, with positive diagonal entries.
b.Show that the QRfactorization of an invertible n×nmatrix is unique. Hint: If, thenthe matrix is both orthogonal and upper triangular, with positive diagonal entries.
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