TITLE: What are the angle brackets in Linear Algebra?
QUESTION [24 upvotes]: In my linear algebra book, they have angle brackets around two different vectors, so it looks like this: $\langle\mathbf{u_2},\mathbf{v}_1\rangle$. They don't use angle brackets to define vectors, but use regular parenthesis instead.
For the Gram-Schmidt process, they define 
$\mathbf{v}_1 = \mathbf{u}_1 = (1,1,1)$
and
$\mathbf{v}_2 = \mathbf{u}_2 = \mathbf{u}_2 - 
\dfrac{\langle\mathbf{u}_2, \mathbf{v}_1\rangle}{\|\mathbf{v}_1\|^2} \mathbf{v}_1$
where $\mathbf{u}_2 = (0,1,1)$
They conclude that that formula is equal to
$(0,1,1) - \dfrac{2}{3}(1,1,1)$.
What operation is the angle brackets to have that result?

REPLY [4 votes]: As Fly by Night said, angle brackets can represent an inner product, as mentioned in wikipedia here: https://en.wikipedia.org/wiki/Dot_product
It can however also be used to contain an ordered set. This can be seen in the ordered set section of the vector notation page of wikipedia: https://en.wikipedia.org/wiki/Vector_notation#Ordered_set_notation
Your example is the first use case, but I wanted to make sure that anyone else looking knows that it could also be used to list vector elements.