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Chapter 4: Q57E (page 185)

Find the kernel and nullity of the transformation TinT(f)=f''-5f'+6fC鈭�

Short Answer

Expert verified

The kernel contains of all the equation of the formae2t+be3tandthenullityis2.

Step by step solution

01

Explanation of the solution

Consider the linear transformation as follows.

T:C鈭�鈫�C鈭�definedasT(f)=f''-5f'+6f.

Since,f(t)鈭�C鈭�

Consider the equation as follows.

Tft=0f'-5f'+6f=0

The characteristic equation as follows

k2-5k+6=0k2-2k-3k+6=0kk-2-3k-2=0k-2k-3=0

02

Simplify further

Simplify as follows.

k-2=0k=2k-3=0k=3

The roots of the equationf'-5f'+6f=0are2and3.

Therefore, the general solution of the equation as follows.

ft=ae2t+be3t

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

ae2t+be3t

Thus, the nullity is 2 with basis is as follows.

e2t,e3t

Hence, the kernel consist of all the function of the form ae2t+be3tand the nullity is 2.

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