1. Gram-Schmidt procedure

I need some help in applying the Gram-Schmidt procedure to the basis {v1; v2; v3} of R^3 ,where where
$\displaystyle v1 = (1; 1; 1) ; v2 = (0; 1; 1) ; v3 = (0; 0; 1);,$in order to obtain an orthogonal basis of R^3 containing (1; 1; 1) and an orthonormal
basis of R^3 containing $\displaystyle (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} ,\frac{1}{\sqrt{3}})$

2. Originally Posted by StefanM
I need some help in applying the Gram-Schmidt procedure to the basis {v1; v2; v3} of R^3 ,where where
$\displaystyle v1 = (1; 1; 1) ; v2 = (0; 1; 1) ; v3 = (0; 0; 1);,$in order to obtain an orthogonal basis of R^3 containing (1; 1; 1) and an orthonormal
basis of R^3 containing $\displaystyle (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} ,\frac{1}{\sqrt{3}})$
1) Let A, B, C be the orthogonal basis of R^3 containing (1, 1, 1).

$\displaystyle A=v_1$

$\displaystyle B=v_2-\frac{A^{T}v_2}{A^TA}A$

$\displaystyle C=v_3-\frac{A^Tv_3}{A^TA}A-\frac{B^Tv_3}{B^TB}B$

2) Divide A, B, C by their lengths to convert to unit vectors.

3. -can you explain to me why you calculated like this?
Divide A, B, C by their lengths to convert to unit vectors.
you are refering to the current A,B,C vectors right?
-$\displaystyle A^{T},B^{T}$ represent the transpose of a vector?

4. We can rewrite in slightly another form.

Let a be a unit vector of A.

The projection of $\displaystyle \ v_2 \$ to a is $\displaystyle \ (a \cdot v_2) \$ and

the vector parallel to a with length $\displaystyle \ (a \cdot v_2) \$ is

$\displaystyle \ (a \cdot v_2) \ a$.

So the vector

$\displaystyle B \ = \ v_2 \ - \ (a \cdot v_2) \ a$

is perpendicular to a.