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There are many useful operations on basis in linear algebra, but a text i'm studying defines those operations only when basis is nonempty.

Matrix representation of linear map is one example. Why don't we define $[T]_{\beta}^{\gamma}$ as $0$ matrix when either $\beta$ or $\gamma$ is empty.

Another is dual basis. Why don't we define $\emptyset ^*$ as $\emptyset$?

Do many theorems fail to hold if these are defined for empty sets? Or are we not defining them simply because we don't need this very much?

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The empty basis is the basis of the zero-dimensional space. Its dual is clearly zerodimensional and has empty basis as well. The matrix with $0$ involved would have no rows at all, not zero rows. The bad thing is that the identity and the zero homomorophism have the same (empty) matriy ... –  Hagen von Eitzen Mar 5 '13 at 7:53
    
@Hagen Do you mean $\emptyset ^*$ is defined to be $\emptyset$ in general? –  Jj- Mar 5 '13 at 8:22

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We do not define $[T]^\gamma_\beta$ to be the zero matrix because the zero matrix already represents a linear map. A zero $n \times m$ matrix represents the zero map from an $m$-dimensional vector space to an $n$-dimensional vector space.

If you allow matrices with zero rows or zero columns then you can certainly represent maps to and from the zero vector space by these matrices, there's nothing logically inconsistent there, in fact this is how some computer programs represent these maps and any theorem that doesn't already require the vector spaces to be non-zero will hold for these matrices. The reason this is not often done outside of computer representations is that it's just not useful.

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Carl de Boor argues that empty matrices would be useful in Matlab: ftp2.cs.wisc.edu/Approx/empty.pdf –  yasmar Mar 5 '13 at 8:39
    
"outside computer representations" –  Jim Mar 5 '13 at 17:57

IMO, you can, and should make definitions and state theorems that account for the degenerate cases, such as the zero vector space spanned by the empty basis.

But one must make correct definitions.

In this case, $[T]_\beta^\gamma$ should still be an $m \times n$ matrix, even when $m$ or $n$ are zero, rather than trying to make it a zero matrix whose dimensions are positive.

You will find that many people avoid treating degenerate cases out of habit and principle, rather than because there is a good or practical reason to do so. Don't read too much into it.

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