Mathematics for Machine Technology
Mathematics for Machine Technology
7th Edition
ISBN: 9781133281450
Author: John C. Peterson, Robert D. Smith
Publisher: Cengage Learning
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Chapter 19, Problem 14A

A 10.00-inch diameter pulley rotates at 160.0 rpm. A belt connects this 10.00-inch diameter pulley with a 6.5-inch diameter pulley. An 8.00-inch diameter pulley is fixed to the same shaft as the 6.50-inch pulley. A belt connects the 8.00-inch pulley with a 3.50-inch diameter pulley. Determine the revolutions per minute of the 3.50-inch diameter pulley. Round the answer to 1 decimal place.

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4. )Consider R5 with dot product and Euclidean norm, a subspace U, and a vector x defined below. Compute the orthogonal projection II, (x) of x onto U, and the distance d(x, U) from x onto U. Include necessary tableau work for linear independence and/or solving system(s) of linear equations. 4 9 2 1 1 -1 U = span[u₁ = 0 , նշ :— 1 , U3 = 3|], x = = 1 0 1 2 -4 2 2 See practice exam 1 #7
Problem 2. Let T be a given linear transformation. 1. Explain in detail why every non-zero scalar multiple of an eigenvector of T is also an eigenvector of T. 2. Show that every non-zero linear combination of two eigenvectors v, w corresponding to the same eigenvalue is also an eigenvector. 3. Prove that a linear combination cv + dw (with c, d 0) of two eigenvectors corresponding to different eigenvalues is never an eigenvector.
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