What are the angle brackets in Linear Algebra?

Solution 1:

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.

Solution 2:

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.