What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:14:28Z http://mathoverflow.net/feeds/question/42410 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42410/what-is-the-dual-concept-to-annihilator-called-and-do-any-linear-algebra-textb What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first? David MJC 2010-10-16T20:07:13Z 2010-11-10T22:58:22Z <p>When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator <code>$\{f\in V^*: f(u)=0\quad \forall u\in U\}$</code> of a subset/space <code>$U$</code> of a vector space <code>$V$</code> is introduced before the dual concept of the "joint kernel"(?) <code>$\{v\in V: f(v)=0\quad \forall f\in W\}$</code> of a subset/space <code>$W$</code> of the dual space <code>$V^*$</code>. The latter notion, despite corresponding to the intuitive idea of the solution space of a homogeneous linear system of equations, is then introduced indirectly by mapping <code>$V$</code> into <code>$V^{**}$</code> and considering the annihilator of the system of equations <code>$W$</code> of <code>$V^*$</code>, which is <code>$\{\phi\in V^{**}: \phi(f)=0\quad \forall f\in W\}$</code>. This seems a pity as many students find the isomorphism of a finite dimensional vector space with its double dual difficult to grasp!</p> <p>Worse, as the "(?)" above suggests, there appears to be no common terminology or notation for the concept dual to annihilator. Possibilities include "(joint) kernel, null/zero space, pre-annihilator, solution space". I'd be grateful for any pointers to textbooks which introduce the concept directly, or any suggestions for terminology and notation.</p> <p><strong>Update</strong>: Many thanks for the suggestions so far. I am still rather surprised and disappointed by the lack of references to elementary linear algebra text books which discuss solution spaces (aka null spaces, joint kernels) before or on the same footing as annihilators. I am unconvinced by the arguments that have so far been made for this omission.</p> <p>Instead it seems to me that the current situation is the result of inertia. The "annihilator" is a term that caught on, and has been carried forward in the absence of a cool name for the dual concept. In this respect I like the invention of the "vanquished".</p> <p>The question is not moot as I am currently lecturing this material. I am going to stick to "solution/null space" and "joint kernel". I will then discuss duality and row rank = column rank without ever mentioning the double dual (the double dual will come later in the course).</p> <p>In preparing this, I noticed the fact that solution spaces and annihilators provide a Galois connection between the posets of subsets/subspaces of a vector space and its dual (that is, solution spaces and annihilators are contravariant adjoint functors between these posets). I've seen discussion of this in the context of annihilators in ring theory, but it seems to me to be at the heart of the concept of duality for vector spaces. I hope this rings some bells among MO readers.</p> <p>Such categorical thinking leads to a fairly clean analysis of solution spaces and annihilators, which has in turn simplified some results subsequently.</p> http://mathoverflow.net/questions/42410/what-is-the-dual-concept-to-annihilator-called-and-do-any-linear-algebra-textb/42415#42415 Answer by Cristi Stoica for What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first? Cristi Stoica 2010-10-16T20:44:50Z 2010-10-17T10:01:31Z <p>Let $V_0\leq V$ be a subspace of the vector space $V$. If $ann(V_0)\leq V^*$ is its annihilator, then the dual of $ann(V_0)$ is isomorphic with the quotient space $V/V_0$:</p> <p>$ann(V_0) \cong \frac{V}{V_0}$</p> <p>This link may be useful: <a href="http://en.wikipedia.org/wiki/Dual_space#Quotient_spaces_and_annihilators" rel="nofollow">http://en.wikipedia.org/wiki/Dual_space#Quotient_spaces_and_annihilators</a></p> <p><strong>Update:</strong></p> <p>Oops, after I added my answer, the page refreshed and I saw that you edited your comment. It seems that my answer does not apply to your question...</p> <p>Anyway, I see no reason to give it a different name than annihilator, even if it annihilates a subspace of $V^*$. It has the same definition.</p> <p><strong>Update 2</strong></p> <p>It may depend on the context:</p> <p>In functional analysis some use the notion of pre-annihilator because it is important to distinguish $V^{**}$ from $V$. Example: Dales, Aiena - Introduction to Banach Algebras, Operators and Harmonic Analysis.</p> <p>When $V^{**}\cong V$ some don't necessarily make the distinction. Example: Marsden and Ratiu - The Breadth of Symplectic and Poisson Geometry</p> <p>But it may be a good practice to specify in what space a subspace is the annihilator of a given subspace: Example: In Automorphic Forms on GL(2), Jacquet and Langlands said: "If $\tilde V_2$ is the annihilator of $V_1$ in $\tilde V$ then $V_1$ is the annihilator of $\tilde V_2$ in $V$.</p> <p>For non-commutative rings, ideals, semigroups, one specifies whether the annihilator is left or right. Examples: G. Gratzer - Universal Algebra Steven G. Krantz (Ed.) - Dictionary of Algebra, Arithmetic, and Trigonometry</p> http://mathoverflow.net/questions/42410/what-is-the-dual-concept-to-annihilator-called-and-do-any-linear-algebra-textb/42417#42417 Answer by Maximiliano Valle for What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first? Maximiliano Valle 2010-10-16T21:12:15Z 2010-10-16T21:27:21Z <p>In the book "Algebra Lineal y Geometría" by Angel Rafael Larotonda, he introduces the concept right after introducing the annihilator of a set $M\in V$ as $M^o$, he introduces the "left" annihilator of a set $F \in V^*$ as $^oF=${$x \in V: f(x)=0 \; \forall \; f \in F$}, the only problem is that its written in spanish. I have also seen the concept introduced directly as pre-annihilator, but alas, also in spanish books.</p>