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Chapter 4: Q32E (page 184)

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

Short Answer

Expert verified

The solution T is not a liner transformation.

Step by step solution

01

Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

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

For all elements f,g of V and k is scalar.

A linear transformation T:V鈫�Wis said to be an isomorphism if and only if ker(T)={0}and im(T)=Wor localid="1659415485656" dim(V)=dim(W).

02

Explanation of the solution        

The given transformation as follows.

Tf=f't+t2, fromP2to P2.

By using the definition of linear transformation as follows.

TA+B=TA+TBTkA=kT(A)

Now, to check the first condition as follows.

Consider the polynomials as follows.

f( x ) and g( x ) fromP2and k is scalar.

Now, simplify as follows.

Tf+g=f+g'+t2=f'+g'+t2鈮�f'+t2+g'+t2Tf+g鈮�Tf+Tg

Thus, the function T is not a linear transformation.

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Most popular questions from this chapter

TRUE OR FALSE?

8. If the kernel of a linear transformation T from P4toP4 is {0}, then T must be an isomorphism.

Find the kernel and nullity of the transformation TinT(M)=[1201]M-M[1201]fromR(2脳2)toR(2脳2)..

Find the image and kernel, rank of the transformationT in T(ft)=f'(t)fromP toP .

Find the basis of each of the spaceV=R3脳2,and determine its dimension.

TRUE OR FALSE?

7. State true or false, the spaceP1 is isomorphic to饾啅 .
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