Can the Graham-Schmidt Process Start with Any Vector?

[rdfloyd]

We learned the GSP yesterday in class, but my teacher said something that seems to conflict with the textbook.

He said that we can set our first vector ($w_{1}$) equal to any vector in the set. However, I keep getting different answers than our textboook when I do this.

My question: Is it possible to set the starting vector equal to any vector in the set? Then, do I have to continue the GSP in the order of the vector subscripts?

For instance, assume that I have vectors $v_{1}, v_{2}$, and $v_{3}$. I want to set $v_{3}$ as my first vector $w_{1}$. So then my list becomes $v_{3}, v_{2}$, and $v_{1}$. Do I now have to do the order backwards, or can I do the order $v_{3}, v_{1}$ (going 1$\rightarrow$ 2 $\rightarrow$ 3), and $v_{2}$, etc.

 

[mathwonk]

if your goals is to get the same answer in your textbook then you need to do it in the same order.

the fact that you get different (correct) answers shows that you can in fact do it in any order, you just get different answers.

if you think about it, it is obvious that there exist very many orthonormal bases for a given subspace.

to see what is going on, try it with (1,0) and (1,1) in both orders.