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Chapter 4: Q65E (page 201)

If T is a linear transformation from V to W and if bothim(T)andker(T)are finite dimensional then V must be finite dimensional.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Define rank-nullity theorem.

If V is finite dimensional then the rank nullity theorem is as follows

dimV= rankT+ nullityT= dimim T+ dimker T

Where rankTis defined as dimim Tand nullityTas dimker T.

02

Determine the dimension of V.

GivenimT and kerTare finite dimensional.

Then by rank nullity theorem as follows.

dimV=rankT+nullityT鈥︹�� (1)

Here, the rank and nullity as follows.

rankT=dimimT=finite

Also,

nullityT=dimkerT=finite

Form equation (1) as follows.

dimV=finite+finitedimV=finite

Hence, V must be finite dimensional.

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