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Chapter 4: Q59E (page 185)

Find the image, kernel, rank, and nullity of the transformationT inT(f(t))=f(7) from P2to R.

Short Answer

Expert verified

The kernel contains of all the functions of the form a(t-7)+bt-72.and the nullity is 2.

Step by step solution

01

Definition of rank of T

Consider the transformation as follows.

T:V鈥�鈫�Wsuch thatim(T)={Tf:f鈭�V} .

If the image of Tis finite dimensional, then dim(imT)is called the rankofT.

02

Explanation of the solution

Consider the linear transformation as follows.

T:P2鈫�Rdefined as Tft=f7.

Consider the equation as follows.

Tft=0f7=0

Since, ft鈭�P2.

Therefore,ft=at-7+bt-72

Thus, the kernel contains all the function of the form as follows.

at-7+bt-72

So, the nullity with the basis as follows.

t-7,t-72

Since, ft=f7.

Andf7鈭�R

Therefore, the image contains all the real number

So, the rank is 1 with the basis 1.

Hence, the kernel contains of all the functions of the form at-7+bt-72.and the nullity is 2 whereas the imagecontains all the real numbers and the rank is 1

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