Re: What's in a name?

"What is the normal mathematical name given to the set of domain points that lead to a range of zero?
Null space?
Or: kernel?

If the question is with regards to an operator (e.g. matrix, integral
operator, or any other operator), then the answer is either "null
space" or "kernel" (as has already been suggested by others). If the
question is with regards to some other mathematical object, you can
ignore the remainder of this post (of course you can ignore it either
way ^__^ )

In particular, suppose an operator A maps vectors from a vector space
X to a vector space Y (often denoted A:X-->Y).
Then the null space N of operator A is defined as
N(A) = { x in X | Ax = 0 }
("the set of all x in X such that Ax=0")

and the range space R of operator A is defined as
R(A) = { y in Y | y=Ax for some x in X }.
("the set of all y in Y such that y=Ax for some x in X")

These two sets, N(A) and R(A), are rightly called "spaces" because
both are "linear subspaces" --- that is, they are both vector spaces
and their elements are vectors such that you can add any two vectors
together, multiply any of them by a scalar to produce another vector
in the set, etc.

The two sets are also important because of their part in the
"fundamental theorem of linear equations":
dim N(A) + dim R(A) = dim X
where dim is the dimension of a subspace and X is a finite dimensional
subspace and A:X-->Y is a linear operator.

The "null space" is also called the "kernel" and the range is also
called the "image" (Bollobas page 28). However, I (howbeit my opinion
not being so significant) would prefer not to use the term "kernel" in
this sense because it is also the name used with the ever popular
integral operators with "kernel" k(x,y) as in
[A f(x)](y) = int_0^1 k(x,y) f(x) dx
The Fourier transform is an example of an integral operator with
kernel
k(t,w) = e^{-iwt} such that
[F f(t)](w) = int_t f(t) e^{-iwt} dt

References:
1. Michel and Herget:
author = {Anthony N. Michel and Charles J. Herget},
title = {Applied Algebra and Functional Analysis},
publisher = {Dover Publications, Inc.},
year = {1993},
isbn = {0-486-67598-X},
oclc = {27770331},
library = {QA154.2.M52 1993},
url = { http://books.google.com/books?vid=ISBN048667598X },
urlw = { http://www.worldcat.org/isbn/048667598X },
urla = { http://www.amazon.com/dp/048667598X },

2. Bollobas:
author = {B\'ela Bollob\'as},
title = {Linear Analysis; an introductory course},
edition = {2},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {1999},
isbn = {0-521-65577-3},
url = { http://books.google.com/books?vid=ISBN0521655773
},
urlw = { http://www.worldcat.org/isbn/0521655773 },
urla = { http://www.amazon.com/dp/0521655773 },

I wish you all the best in your study,
Dan Greenhoe

.

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