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Chapter 4: Q27E (page 200)

State true or false, if the image of a linear transformation T from P to P is all P, thenmust be an isomorphism.

Short Answer

Expert verified

The given statement is True

Step by step solution

01

Determine the dimension of kernel of T.

Consider a linear transformation T from P to P such that the image of T is whole P.

Theorem: Consider a linear transformation T defined fromT:V鈭�W then the transformation Tis an isomorphism if and only iflocalid="1659436257184" dim(Ker(T))=0where Ker(T)={f(x)鈭�P:T{f(x)}=0}anddim(Ker(T))={f(x)鈭�P:T{f(x)}=0impliesf(x)=0}.

A kernel of a function Tis definedKer(T)={f(x)鈭�P:T{f(x)}=0}.

02

Determine if the statement is True of False

As the image of linear transformation T is whole P implies dimKer(T)=0.

By the theorem, the linear transformation T is isomorphism.

Hence, the statement is true.

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