I am reading "Coding the Matrix" by Philip N. Klein.
There are the following definitions in this book:
Definition 6.5.7:
For a subspace $\mathcal{V}$ of $\mathbf{F}^n$, the annihilator of $\mathcal{V}$, written $\mathcal{V}^o$, is $$\mathcal{V}^o=\{\mathbf{u}\in\mathbf{F}^n : \mathbf{u}\cdot\mathbf{v}=0 \text{ for every vector } \mathbf{v}\in\mathcal{V}\}.$$
Definition 9.6.1:
Let $\mathcal{W}$ be a vector space over the reals, and let $\mathcal{U}$ be a subspace of $\mathcal{W}$. The orthogonal complement of $\mathcal{U}$ with respect to $\mathcal{W}$ is defined to be the set $\mathcal{V}$ such that $$\mathcal{V}=\{\mathbf{w}\in\mathcal{W} : \mathbf{w} \text{ is orthogonal to every vector in }\mathcal{U} \}.$$
In this book, $\mathcal{V}$ is called a vector space if and only if $\mathcal{V}$ is a subspace of $\mathbf{F}^n$.
What is the difference between the annihilator of a vector space and the orthogonal complement in this book?
I think the annihilator of $\mathcal{V}$ is the orthogonal complement of $\mathcal{V}$ with respect to $\mathbf{F}^n$.
Am I right?
Maybe a little late, but we usually think about a bit more abstract vector spaces to have a clearer understanding instead of just working on $\mathbb{F}^n$. If we let $V$, be a vector space (with finite dimension, not really sure this needs it just to be sure it works out), we want to think of the set of linear functions that map from $V$ to $\mathbb{F}$ usually denoted $\mathcal{L}(V,\mathbb{F})$. If we take an element from this set, we will be taking a function.
Let $\phi$ be an element of $\mathcal{L}(V, \mathbb{F})$, so it will be linear and it will also transport the vectors to your field. When we take the annihilator of a subset we are actually talking about a subspace in this space of functions. For example let $U$ be a subset of $V$, when we take the annihilator of $U$ denoted by $U^0$ we are actually looking for the set of functions that there null space contains $U$.
The orthogonal complement of that set $U$, usually it is defined the way you have it, but in more abstract cases we actually measure the angle via de dot product, so actually de definition of the orthogonal complement is the set of vectors such that their dot product is zero with every element in $U$.
There's actually a connection between the two, spaces that I was looking into when I stumbled into your question, but I would credit it to the Riesz Reresentition Theorem. Where every element in $\mathcal{L}(U, \mathbb{F})$ has a corresponding vector $v$, such that $\phi(u) = u \cdot v$, for all elements in $V$.