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Chapter 4: Q71E (page 185)

Does there exist a polynomial f(t) of degree 鈮�4such that role="math" localid="1659756086073" f(2)=3f(3)=5,f(5)=7,f(7)=11andf(11)=2?so, how many such polynomials are there? Hint: Use Exercise 70.

Short Answer

Expert verified

There exist a polynomial of degree such that ,f(2)=3f(3)=5,f(5)=7,f(7)=11andf(11)=2

Step by step solution

01

Definition of Linear transformation

Consider two linear spaces Vand W. A function Tfrom Vto Wis called linear transformation if

(i)T(f+g)=T(f)+T(g)(ii)T(kf)=kT(f)

for all elements f and g of Vand for all scalars k.

02

Definition of Isomorphism.

An invertible linear transformation Tis called an isomorphism.

03

Determination of a polynomial of degree less than 4

Consider the real numbers 2,3,5,7,11.

The linear transformation T:P4鈫�鈩�5is given by,

Tft=f2f3f5f7f11

The numbers all are distinct.

Since a polynomial of degree less or equal than n has at most n roots, we have that there exists no polynomial of degree less than or equal to n other than the zero polynomial.

Thus, kerT=0imT=鈩�5.

Since, T is an isomorphism, there exists a unique ft鈭�P4such that forT-1v=ftforanyv鈭�鈩�5 .

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Most popular questions from this chapter

T denotes the space of infinity sequence of real numbers,T(f(t))=f(t2)from PtoP.

Find the basis of all Pn, and determine its dimension.

In Exercise 72through 74, let Znbe the set of all polynomials of degree鈮�nsuch that f(0) = 0.

73. Is the linear transformationT(ft)=鈭�0tf(x)dx an isomorphism fromPn-1 toZn?

Find the transformation is linear and determine whether the transformation is an isomorphism.

Let V be the space of all infinite sequences of real numbers. See Example 5. Which of the subsets of given in Exercises 12 through 15 are subspaces of V ? The arithmetic sequences [i.e., sequences of the form(a,a+k,a+2k,a+3k,...) , for some constants and K .
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