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Chapter 5: Q6E (page 233)
If the matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?-B.
The Matrix -B is orthogonal.
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Consider a linear transformationL from to that preserves length. What can you say about the kernel of L? What is the dimension of the image? What can you say about the relationship between n and m? If Ais the matrix of L, What can you say about the columns of A? What is? What about? Illustrate your answer with an example where m=2and n=3.
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.)
Complete the proof of Theorem 5.1.4: Orthogonal projection is linear transformation.
Consider a subspace Vof and a vector in. Let . What is the relationship between the following quantities?
and
Consider a symmetric invertible n脳nmatrix Awhich admits an LDU-factorization A=LDU. See Exercises 90, 93, and 94 of Section 2.4. Recall that this factorization is unique. See Exercise 2.4.94. Show that
(This is sometimes called the - factorizationof a symmetric matrix A.)
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