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I'm wondering what symbol is used for the kernel and the range of a linear transformation. I've seen them being written as such:

$\ker(T)$

$\mathcal{R}(T)$

Are these the "correct" symbols for the kernel and range, or do they differ between different mathematicians? Are there any that are used more frequently? This is being a bit pedantic, but I'm just curious.

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    $\begingroup$ I've also seen $\text{im} (T) $ used for range. $\endgroup$ – Mark Dec 10 '16 at 20:45
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    $\begingroup$ For the former, I've also seen $\mathcal{N}(T)$; for the later, it's right. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Dec 10 '16 at 20:46
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    $\begingroup$ Usually, $\ker T$ is paired with $\operatorname{im} T$, and $\mathcal{R}(T)$ is paired with $\mathcal{N}(T)$. $\endgroup$ – Daniel Fischer Dec 10 '16 at 20:51
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    $\begingroup$ @SkeletonBow Feel free to answer your own question with what you have learned! It is highly encouraged on this site. $\endgroup$ – Ken Duna Dec 10 '16 at 21:24
  • $\begingroup$ Thanks for the suggestion! $\endgroup$ – Skeleton Bow Dec 10 '16 at 21:42
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Daniel Fisher gave a good comment which I'm putting here:

Both of the notations are correct, but usually those two are not used together because of consistency. Here are the "pairs" that are usually used together for the kernel and the range:

$\ker T$ and $\operatorname{im} T$

$\mathcal{N}(T)$ and $\mathcal{R}(T)$

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