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Chapter 7: Q21E (page 324)

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 inR3.

Short Answer

Expert verified

Eigen vectors are all vectorsv=R3

Eigen value will be位=5

Eigen basis ise1鈫�,e2鈫�,e3鈫�

It is diagonalizable

Step by step solution

01

Definition of eigenbasis

An eigenbasis consist of eigen value of a matrix .

02

Note the given data

Clearly TV = 5V

03

Calculation scaling by 5 in R3

Every vector v=R3is an eigen vector of A, with the corresponding eigen value will be 位=5

Hence the canonical basis e1鈫�,e2鈫�,e3鈫�for v=R3is an eigen basis.

Thus, the matrix of T is A=5l3which is already diagonal.

So T is diagonalisable

Hence the solutions are

Eigen vectors are all vectors

Eigen value will be位=5

Eigen basis ise1鈫�,e2鈫�,e3鈫�

T is diagonalisable.

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Most popular questions from this chapter

In all parts of this problem, let V be the linear space of all 2 脳 2 matrices for which [12]is an eigenvector.

(a) Find a basis of V and thus determine the dimension of V.

(b) Consider the linear transformation T (A) = A[12] from V to R2. 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[13] from V to R2. Find a basis of the image of L and a basis of the kernel of L. Determine the rank of L.

(a) Give an example of a 3 脳 3 matrix A with as many nonzero entries as possible such that both span(e鈫�1) and span(e鈫�1,e鈫�2) are A-invariant subspaces of . See Exercise 65.

(b) Consider the linear space Vof all 3 脳 3 matrices A such that both span (e鈫�1) and span (e鈫�1,e鈫�2) are A-invariant subspaces of R3 . Describe the space V (the matrices in V 鈥渉ave a name鈥�), and determine the dimension of V.

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

x炉(t)=[wtm(t]]

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" x炉(t++1)=Ax鈫�(t). 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?

if A is a 2脳2matrix with t r A = 5and det A = - 14what are the eigenvalues of A?

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[51-521082-7]
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