# Does “nullity” have a potentially conflicting or confusing usage?

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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Perhaps if he used nullity, he would feel obliged to call the dimension of the column and row spaces the columnity and rowity, respectively. – Gerry Myerson Jul 31 '12 at 12:25
I think it is sort of a silly term myself. The dimension of the kernel should have a more dignified name; it is a very important dimension. – Qiaochu Yuan Jul 31 '12 at 13:13
Are you sure that the author doesn't use Nul $A$ as shorthand for the nullspace of $A$? If so, then the statement seems reasonable. – shoda Aug 30 '12 at 14:43
The word “nullity” is best applied to individual humans. – Lubin Sep 29 '12 at 19:16

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 $$\operatorname{rk} f+ \dim\operatorname{coker} f = \dim W.$$
I don't like "rank" a lot either. If I were going to say this, I would say "Dimension theorem: $\dim{Im(f)}+\dim{ker(f)}=n$. – rschwieb Jul 31 '12 at 14:12