Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the image and kernel of the transformation $T$ in $T (x_{0}, x_{1}, x_{2}, x_{3}, . . .) = (0, x_{0}, x_{2}, x_{4})$ from $V$ to $V$ .

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Explanation of the solution

Consider the sequence as follows.

$T (x_{0}, x_{1}, x_{2}, x_{3}, . . .) = T (x_{0}, x_{2}, x_{4})$

The image of the sequence is as follows.

$(0, x_{0}, x_{2}, x_{4}, . . .)$

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

Step2: Find kernel of the sequence

Consider the sequence as follows.

$T (x_{0}, x_{1}, x_{2}, x_{3}, . . .) = T (x_{0}, x_{2}, x_{4})$

Now, kernel of the sequence as follows.

$T (0, x_{0}, x_{1}, x_{2}, 0, x_{2}, 0, x_{3}, 0, . . .) = (0, 0, 0, 0, . . .) x_{0} = x_{1} = x_{2} = . . . = 0$

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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Linear Algebra With Applications

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Short Answer

Find the image and kernel of the transformation $T$ in $T (x_{0}, x_{1}, x_{2}, x_{3}, . . .) = (0, x_{0}, x_{2}, x_{4})$ from $V$ to $V$ .

Step by Step Solution

Step1: Explanation of the solution

Step2: Find kernel of the sequence

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Find the image and kernel of the transformation $T$ in $T (x_{0}, x_{1}, x_{2}, x_{3}, . . .) = (0, x_{0}, x_{2}, x_{4})$ from $V$ to $V$ .