Gram-Schmidt Procedure: Step 2 - Calculating Inner Product

[iScience]

(i just need help with step 2)

step 1.) normallize anyone of the given vectors (vector divided by the magnitude)

step 2.) |e₂> - <e₁'|e₂>|e₁'>

where e₁' is the normallized e₁.so in step 2, <e₁'|e₂> is the inner product which would yield... the component (a scalar) of |e₁'> along |e₂>?

 

[wotanub]

It makes a lot more sense if you draw a picture and think about just regular two-space. Draw two vectors that are not orthogonal on a sheet of paper and think about what you would do if you were trying to find an orthonormal basis.

I don't really want to give it away since it's a great exercise, but since you posted I guess you might be stumped, so just read the rest if you want a hint.

First, you could pick either then scale it to length 1. Then you'd look at the other vector and find its components in terms of your first vector and a vector (that you just draw on the sheet) orthogonal to the first one. Then you'd use the components and vector addition rules to write an expression for that second orthogonal vector and so on.

It gets more involved as you do more and more dimensions, but there is an algorithm to be followed.

Wikipedia has this nice picture showing how it's done in 3D...