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Chapter 4: Q46E (page 200)

Question: If T is a linear transformation from P6toP6that transformtk into a polynomial of degreek(fork=0,1,...,6) (for ) then T must be an isomorphism鈥�.

Short Answer

Expert verified

The solution is the statement is false.

Step by step solution

01

Determine the linear transformation

Consider the transformation as follows.

T:P6鈫�P6can be defined by as follows.

T1=0andT(tk)=tkfork=1,2,3,...,6.

02

Determine the statement is True or False

Then T is a linear transformation such thattk into a polynomial of degree for .

k = 1,2,3,...,6.

But here,T1=0 implies that 1鈭�kerT.

Hence, T is not injective.

Therefore, T is not an isomorphism.

Thus, the given statement 鈥淚f T is a linear transformationT:P6鈫�P6 that transformstk into a polynomial of degree k then T must be an isomorphism is false.

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Most popular questions from this chapter

Question: If T is linear transformation from V to V, then {f鈭�V:Tf=f} must be a subspace of V.

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