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

$\mathbf{v}_1 = \mathbf{u}_1 = (1,1,1)$


$\mathbf{v}_2 = \mathbf{u}_2 = \mathbf{u}_2 - \dfrac{\langle\mathbf{u}_2, \mathbf{v}_1\rangle}{\|\mathbf{v}_1\|^2} \mathbf{v}_1$

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?

  • 1
    $\begingroup$ It's the inner product, probably the standard inner product $\langle u,v\rangle = \sum u_i\cdot v_i$ (if real, conjugate one factor if complex). $\endgroup$ Commented Nov 17, 2013 at 20:20
  • 4
    $\begingroup$ There should be a definition for the symbol in the textbook. $\endgroup$
    – egreg
    Commented Nov 17, 2013 at 20:24
  • $\begingroup$ I'm sure it's somewhere. I must have missed it when it was introduced. $\endgroup$
    – jfa
    Commented Nov 17, 2013 at 20:27
  • $\begingroup$ @DanielFischer if you want to go ahead and make that an answer since you were the first one to propose it, I can make that the correct answer. $\endgroup$
    – jfa
    Commented Nov 17, 2013 at 20:50
  • $\begingroup$ @Jack Nah, you got a pretty good answer, I think. Unless you're not satisfied with that, I see no reason to add another one. $\endgroup$ Commented Nov 17, 2013 at 20:55

2 Answers 2


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.


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .