I'm trying to understand a paper from Brakerski and Vaikuntanathan. In it, they use the notation $\langle a,b \rangle$ but they don't explain what that means.
The answers on this question say that it's just another way to write a tuple, but I don't think that's right in this case since the authors use the regular $(a,b)$ notation in the rest of the paper to denote a tuple.
Does anyone know what else $\langle a,b \rangle$ could mean?
In mathematics, $\langle a,b \rangle$ represents the inner product of two vectors and this is a generalization of the dot product. It is one of the multiplications of the vectors in a Vector space and the result is a scalar.
An inner product satisfies these properties
$\langle c+a,b\rangle =\langle c,a\rangle +\langle a,b\rangle $.
$\langle \alpha a,b\rangle = \alpha\langle a,b\rangle$ where $\alpha$ is a scalar.
$\langle a,b\rangle =\langle b,a\rangle$.
$\langle a,a\rangle = 0 $ and equal if and only if $v=0$.
As noted in the comments by @levgeni, if the field is $\mathbb{F}_2$ then
$$\langle (1,1),(1,1) \rangle = 1 \cdot 1 + 1 \cdot 1 = 2 = 0$$ and $$\langle (1,1),(1,0) \rangle = 1 \cdot 1 + 1 \cdot 0 = 1 = 1$$ therefore, the weaker condition is required.
The inner product definition changes according to the vector space. In the context of the paper, let
$$a =(a_1,\ldots,a_n)$$ $$b =(b_1,\ldots,b_n)$$
be two n-dimensional vectors then;
$$\langle a, b \rangle = a_1 \cdot b_1 + \cdots + a_n \cdot b_n $$
