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Chapter 4: Q58E (page 185)

Find the image and kernel of the transformation TinT(x0,x1,x2,x3,...)=(0,x0,x2,x4) from Vto V.

Short Answer

Expert verified

The solution is the image consist of all infinite sequence with the initial element as 0 and the kernel contains zero sequence only.

Step by step solution

01

Explanation of the solution

Consider the sequence as follows.

T(x0,x1,x2,x3,...)=Tx0,x2,x4

The image of the sequence is as follows.

0,x0,x2,x4,...

Therefore, the image consist of all infinite sequence whose initial element is 0.

02

Find kernel of the sequence

Consider the sequence as follows.

T(x0,x1,x2,x3,...)=Tx0,x2,x4

Now, kernel of the sequence as follows.

T0,x0,x1,x2,0,x2,0,x3,0,...=0,0,0,0,...x0=x1=x2=...=0

Thus, the kernel contains zero sequence only.

Hence, the image consist of all the infinite sequence with initial element as 0 whereas the kernel contains zero sequence only.

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