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In Linear Algebra and Its Applications, David Lay writes, "the dimension of the null space is sometimes called the nullity of A, though we will not use the term." He then goes on to specify "The Rank Theorem" as "rank A + dim Nul A = n" instead of calling it the the rank-nullity theorem and just writing "rank A + nullity A = n".

Naturally, I wonder why he goes out of his way to avoid using the term "nullity." Maybe someone here can shed light....

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    I would have thought that nullity is perfectly ok. $B$ut at least four people whose opinions I regard highly apparently disagree. Making a mental note here. My excuse is that I have not encountered the word nullity in a context other than linear algebra, so the word is "just a word" for me. Apparently the word is not loaded with any undignified overtones for an ESL-user like me.2013-04-08

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While choices of terminology is often a matter of taste (I would not know why the author should prefer to say $\operatorname{Nul}A$ instead of $\ker A$), there is at least a mathematical reason why "rank" is more important than "nullity": is it connected to the matrix/linear map in a way where source and destination spaces are treated on equal footing, while the nullity is uniquely attached to the source space. This is why the rank can be defined in an equivalent manner as the row rank or the column rank, or in a neutral way as the size of the largest non-vanishing minor or the smallest dimension of an intermediate space through which the linear map can be factored (decomposition rank). No such versatility exists for the nullity, it is just the dimension of the kernel inside the source space, and cannot be related in any way to the destination space. A notion analogous to the nullity at the destination side is the codimension of the image in the destination space (that is, the dimension of the cokernel); it measures the failure to be surjective, and it is different from the nullity (which measures the failure to be injective) for rectangular matrices. There is a (rather obvious) analogue to the rank-nullity theorem that says that for linear $f:V\to W$ one has $ \operatorname{rk} f+ \dim\operatorname{coker} f = \dim W. $

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Maybe he just doesn't like the idea of using jargon for the dimension of a specific subspace? It's not a particularly useful piece of jargon.

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    @rschwieb Yea, if I didn't like the term "nullity", I would've stated the theorem (in terms of the matrix A) as "dim Col A + dim Nul A = n". But then this would still be favouritism -- of 'Col A' over 'Row A'. Haha2012-08-01