Gram-Schmidt orthonormal basis

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.

The Attempt at a Solution

matt grime
Homework Helper
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.

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
Homework Helper

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

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.