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). $\endgroup$ Nov 17 '13 at 20:20
  • 4
    $\begingroup$ There should be a definition for the symbol in the textbook. $\endgroup$
    – egreg
    Nov 17 '13 at 20:24
  • $\begingroup$ I'm sure it's somewhere. I must have missed it when it was introduced. $\endgroup$
    – JFA
    Nov 17 '13 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
    Nov 17 '13 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. $\endgroup$ Nov 17 '13 at 20:55

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.


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