Gramschmidt orthonormalization : gramschmidt

### 2.49.4 Gramschmidt orthonormalization : `gramschmidt`

`gramschmidt` takes one or two arguments :

- a matrix viewed as a list of row vectors, the scalar product being the canonical scalar product, or
- a list of elements that is a basis of a vector subspace, and a function that defines a scalar product on this vector space.

`gramschmidt` returns an orthonormal basis for this scalar product.
Input :

`normal(gramschmidt([[1,1,1],[0,0,1],[0,1,0]]))`

Or input :

`normal(gramschmidt([[1,1,1],[0,0,1],[0,1,0]],dot))`

Output :

`[[(sqrt(3))/3,(sqrt(3))/3,(sqrt(3))/3],[(-(sqrt(6)))/6,(-(sqrt(6)))/6,(sqrt(6))/3],[(-(sqrt(2)))/2,(sqrt(2))/2,0]]`

Example
We define a scalar product on the vector space of polynomials by:

P.Q= | ∫ |
1
−1
| P(x).Q(x)dx

Input :

`gramschmidt([1,1+x],(p,q)->integrate(p*q,x,-1,1))`

Or define the function `p_scal`, input :
`p_scal(p,q):=integrate(p*q,x,-1,1)`
then input :

`gramschmidt([1,1+x],p_scal)`

Output :

`[1/(sqrt(2)),(1+x-1)/sqrt(2/3)]`