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Chapter 4: Q66E (page 185)

Find the kernel and nullity of the transformation T inT(f)=f-f fromc鈭� to c鈭�.

Short Answer

Expert verified

The kernel contains of all the equation of the formaet and the nullity is 1.

Step by step solution

01

Explanation of the solution

Consider the linear transformation as follows.

T:C鈭�鈫�C鈭�defined as T(f)=f-f'.

Since, f(t)鈭�C鈭�.

Consider the equation as follows.

Tft=0f-f'=0

02

Find the characteristic equation

The characteristic equation as follows.

1-k=0k=1

The roots of the equationf-f'=0 are 1.

Therefore, the general solution of the equation as follows.

f(t)=aet

The kernel consist of all the function in the form as follows.

aet

Thus, the nullity is 1.

Hence, the kernel consist of all the function of the form role="math" localid="1659424020947" aetand the nullity is 1.

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