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Is there a linear transformation on $\mathbb R^3$ (the usual 3-dimensional vector space) whose image and kernel are the same?

How it can be same if their addition must be 3?

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No.

Suppose Image=Kernel $\implies$ $r=\text{dim (Image)}=\text{dim (Kernel)}$. Then by Rank Nullity Theorem, $r+r=3$ which has non integer solutions, hence a contradiction.

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  • $\begingroup$ Thank you I also thought that $\endgroup$
    – subhanceo
    Commented Jun 24, 2020 at 15:48

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