Man, I am really lost with what you are asking now. I think you need to review some of your definitions, it is getting to the point that these posts are becoming incoherent to me.

You do not find a basis for a matrix.

A matrix represents a linear transformation, a basis is for a vector space, not a function. Matrices have ranks, it does not make sense to talk about the rank of a row.

The dual space to does not have a "standard" basis. It is not canonically isomorphic to . It is isomorphic, but only in terms of a basis. If you give me a basis for , I will give you a basis for .

I already wrote out the basis for the dual space in a previous post on here titled "bassis(I can't type apparently) for the Dual Space".

Look at that. If you want the basis for the dual space for subordinant to the standard basis, then

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and this yields the basis for

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