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I know that when we have something like $\bf u$, then the arbitrary element $i$ will be denoted as $u_{i}$.

Now suppose we have vectors $\{\mathbf {v_{1}, v_{2}, \cdots, v_{n}} \}$

I want to consider, say, third element of the fifth vector.

How do I denote it?

Is it $(\mathbf{v_{5}})_{3}$ or $\mathbf{v_{}}_{5_{3}}$? Or are there better options available?

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    $\begingroup$ Depending on the context, you may want $v_{5,3}, v_5(3), v_5[3], v_5^3$ (though you may make some people irritated, no matter which you choose). Just make sure you are clear and consistent with your notation, and if you find you later have clashes of notation, be very careful to specify which is which. $\endgroup$ – Brian Moehring Sep 20 at 5:39
  • $\begingroup$ Yes, ${V^_5)_3$ is correct. $\endgroup$ – Dr Zafar Ahmed DSc Sep 20 at 5:54
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You can technically do whatever you want as long as it’s comprehensible (actually, it doesn’t even need to be comprehensible, but if it isn’t, nobody will care—just look at some of Galois’s manuscripts!) but what you might find useful is $v_{i,j}$ which may denote the $i$-th component of the $j$-th vector, or vice-versa.

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    $\begingroup$ I like this notation, as it also is in parallel with notation for matrix elements. $\endgroup$ – Don Thousand Sep 20 at 15:22
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When I have to manipulate indices of many vectors, I like calling them $\{v^{(1)},...,v^{(m)}\}$ and labeling elements of those vectors $v^{(i)}_j$ because I prefer to keep vector indices downstairs when I can. You can also drop the parantheses. Whatever makes most sense to you is good, as long as other people can understand it as well.

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