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Q58E

Expert-verifiedFound in: Page 185

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Find the image and kernel of the transformation ${\mathbf{T}}$in${\mathbf{T}}{\left({x}_{0},{x}_{1},{x}_{2},{x}_{3},...\right)}{\mathbf{=}}{\left(0,{x}_{0},{x}_{2},{x}_{4}\right)}$ from ${\mathbf{V}}$to ${\mathbf{V}}$.**

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

Consider the sequence as follows.

$\mathrm{T}({\mathrm{x}}_{0},{\mathrm{x}}_{1},{\mathrm{x}}_{2},{\mathrm{x}}_{3},...)=\mathrm{T}\left({\mathrm{x}}_{0},{\mathrm{x}}_{2},{\mathrm{x}}_{4}\right)$

The image of the sequence is as follows.

$\left(0,{\mathrm{x}}_{0},{\mathrm{x}}_{2},{\mathrm{x}}_{4},...\right)$

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

Consider the sequence as follows.

$\mathrm{T}({\mathrm{x}}_{0},{\mathrm{x}}_{1},{\mathrm{x}}_{2},{\mathrm{x}}_{3},...)=\mathrm{T}\left({\mathrm{x}}_{0},{\mathrm{x}}_{2},{\mathrm{x}}_{4}\right)$

Now, kernel of the sequence as follows.

$T\left(0,{x}_{0},{x}_{1},{x}_{2},0,{x}_{2},0,{x}_{3},0,...\right)=\left(0,0,0,0,...\right)\phantom{\rule{0ex}{0ex}}{x}_{0}={x}_{1}={x}_{2}=...=0$

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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