What are the angle brackets in Linear Algebra? 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? 
 A: 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.
A: The angled brackets represent an inner product. The best known one is the scalar product or the dot product. If ${\bf u} = (u_1,u_2,u_3)$ and ${\bf v} = (v_1,v_2,v_3)$, then the dot product is given by
$$\langle {\bf u},{\bf v} \rangle = u_1v_1 + u_2v_2+u_3v_3$$
It has many useful properties. First $\langle {\bf u},{\bf u} \rangle = \|{\bf u}\|^2$, and second if ${\bf u}$ and ${\bf v}$ are both non-zero then $\langle {\bf u},{\bf v}\rangle = 0$ if and only if ${\bf u}$ and ${\bf v}$ are orthogonal. In general:
$$\langle {\bf u}, {\bf v} \rangle = \|{\bf u}\| \|{\bf v}\| \cos\theta$$
where $\theta$ is the angle between ${\bf u}$ and ${\bf v}$. This idea can be generalised.  Notice that $\langle {\bf u},{\bf v} \rangle = {\bf u}E{\bf v}^{\top}$, where $E$ is the 3-by-3 identity matrix and ${\bf u}$ and ${\bf v}$ are being thought of as 1-by-3 matrices. For any 3-by-3 matrix, say $M$, we can define $\langle {\bf u},{\bf v}\rangle_M := {\bf u}M{\bf v}^{\top}$. Different matrices gives rise to different $\langle {\bf u},{\bf v}\rangle_M$. We usually assume that $M$ is a positive definite matrix.
