Gram-Schmidt orthonormal basis

[Spectos]

Homework Statement

Let R^3 have the inner product <u, v> = u₁v₁ + 2u₂v₂ + 3u₃v₃. Use the Gram-Schmidt process to convert u₁=(1,1,1), u₂ = (1,1,0), u₃ = (1,0,0) into a normal orthonormal basis

Homework Equations

I know the process for the orthonomoral converasion. I have no problem with that. I hope I can skip typing those equations out.

The Attempt at a Solution

My confusion is how to use the equation for the dot product. I am thinking that I need to do the Gram-Schmidt process then somehow put the result into the dot product equation.

If I phrased anything wrong or left out something pertinent I apologize this is my first post. So I tried to follow the rules.

 

[matt grime]

If you know the formula, then what is the problem? You have the vectors and the inner product, so you just plug and chug.
Let's start you off - we need to normalize u_1, i.e replace u_1 with

u_1/ sqrt(<u_1,u_1>) = u_1/sqrt(6).

Note that this is the least sensible way to do this. It would be better to use u_3 as the first element to orthonormalize, since that satisfies <u_3,u_3> = 1, i.e. it is already a unit vector. But I presume that the book/class wants you to go through them in order and normalize u_1 to get v_1, then to work out u_2 - <u_2,u_1> and normalize that, etc.

 

[Spectos]

Then maybe I am reading too far into. I am just not sure how the dot product equation in the beginning of question is utilized. As in after I convert $$\vec{u_1}$$, $$\vec{u_2}$$, $$\vec{u_3}$$. So do I just plug in the new normalized $$\vec{u_1}$$, $$\vec{u_2}$$, $$\vec{u_3}$$ and then use $$\vec{u_1}$$ = $$\vec{v_1}$$, $$\vec{u_2}$$=$$\vec{v_2}$$, $$\vec{u_3}$$=$$\vec{v_3}$$ into <u, v> = u₁v₁ + 2u₂v₂ + 3u₃v₃.

 

[Cyosis]

The general Gramm-Schmidt formula reads:

$$v_j=a_j-\left( \frac{\langle a_j,v_1 \rangle}{\langle v_1,v_1 \rangle}v_1+\frac{\langle a_j,v_2 \rangle}{\langle v_2,v_2 \rangle}v_2+\ldots+\frac{\langle a_j,v_{j-1} \rangle}{\langle v_{j-1},v_{j-1} \rangle}v_{j-1} \right)$$.

This should make it pretty obvious where to use the dot product, so I am curious what kind of formula were you thinking of?

 

[HallsofIvy]

For example, to "normalize" the first vector, you need to divide by its length. And its length is $||v||= \sqrt{v\cdot v}$. Because of the definition of dot product, $(1,1,1)\cdot(1,1,1)= 1(1)+ 2(1)(1)+ 3(1)(1)= 6$. That's where matt grime got the $\sqrt{6}$ he divided the first vector by.

 

[Spectos]

Ok, I was forgetting/missing to use the equation when getting the norm or mag to normalize at the end. I think I am good. Thanks for all the quick help.

 

[matt grime]

You don't just use it at the end to normalize - you use it at every occasion that you need to take an inner product.