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Chapter 7: Q53E (page 384)
If v鈫抜s an eigenvector of A, then v鈫抦ust be in the kernel of Aor in the image ofA.
The given statement is true.
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For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology
consider an eigenvalue of anmatrix A. we are told that the algebraic multiplicity of exceeds 1.Show that(i.e.., the derivative of the characteristic polynomial of A vanishes are).
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?
Consider the linear space of allmatrices for which all the vectorsare eigenvectors. Describe the space(the matrices in"have a name"), and determine the dimension of.
In all parts of this problem, let V be the linear space of all 2 脳 2 matrices for which is an eigenvector.
(a) Find a basis of V and thus determine the dimension of V.
(b) Consider the linear transformation T (A) = A from V to . Find a basis of the image of Tand a basis of the kernel of T. Determine the rank of T .
(c) Consider the linear transformation L(A) = A from V to . Find a basis of the image of L and a basis of the kernel of L. Determine the rank of L.
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