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Chapter 7: Q46E (page 358)
Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.
from the space V of infinite sequences into V. (We drop every other term.)
The eigenvalues and eigenvectors for the given linear equation is,
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Arguing geometrically, find all eigen vectors and eigen values of the linear transformations. In each case, find an eigen basis if you can, and thus determine whether the given transformation is diagonalizable.
Scaling by 5 in.
24: Find all eigenvalues of the positive transition matrix
See Definitions 2.1.4 and 2.3.10.
Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.
Orthogonal projection onto a line L in.
28 : Consider the isolated Swiss town of Andelfingen, inhabited by 1,200 families. Each family takes a weekly shopping trip to the only grocery store in town, run by Mr. and Mrs. Wipf, until the day when a new, fancier (and cheaper) chain store, Migros, opens its doors. It is not expected that everybody will immediately run to the new store, but we do anticipate that 20% of those shopping at Wipf鈥檚 each week switch to Migros the following week. Some people who do switch miss the personal service (and the gossip) and switch back: We expect that 10% of those shopping at Migros each week go to Wipf鈥檚 the following week. The state of this town (as far as grocery shopping is concerned) can be represented by the vector
where w(t) and m(t) are the numbers of families shopping at Wipf鈥檚 and at Migros, respectively, t weeks after Migros opens. Suppose w(0) = 1,200 and m(0) = 0.
a. Find a 2 脳 2 matrix A such that role="math" localid="1659586084144" . Verify that A is a positive transition matrix. See Exercise 25.
b. How many families will shop at each store after t weeks? Give closed formulas. c. The Wipfs expect that they must close down when they have less than 250 customers a week. When does that happen?
Give an example of a matrixAof rank 1 that fails to be diagonalizable.
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