# Calculating an Orthonormal Basis

Hi, I'm still quite inexperienced with Sage at the moment, so forgive me if this is a basic issue. I am trying to produce an orthonormal basis, I have created the orthogonal complement to my original basis by taking its left nullspace using kernel() I now want to use gram_schmidt() to produce a normalised version. I am unsure however what ring my input matrix should be over. QQbar won't work for me, but I am worried that the RDF result I get will not retain the linear independence that I need, as it says in the documentation for gram_schmidt() that under RDF, *"no attempt is made to recognize linear dependence with approximate calculations"*
My input matrix is the following;

```
[ 0 -1 0 0 1 0 0 0 0 0 0 0]
[-1/2 0 0 -1/2 0 0 1 0 0 0 0 0]
[-1/2 -1 0 1/2 0 0 0 1 0 0 0 0]
[-1/2 0 0 -1/2 0 0 0 0 0 1 0 0]
[ 1/2 -1 0 -1/2 0 0 0 0 0 0 1 0]
[ 0 0 -1 0 0 -1 0 0 1 0 0 1]
```

if I change the ring of this matrix to RDF, gram_schmidt() runs but the inexact entries of -0.0 and what is clearly 1/sqrt(2) are not so useful.

```
[ 0.0 -0.408248290464 0.353553390593
-0.288675134595 0.353553390593 -1.32686526214e-17]
[ 0.707106781187 -8.67632788532e-17 0.353553390593
-1.40946282423e-17 -0.353553390593 -1.72861506093e-17]
[ -0.0 0.0 0.0
1.9952420559e-17 3.41114374126e-17 -0.5]
[ -0.0 -0.408248290464 -0.353553390593
-0.288675134595 -0.353553390593 4.70626515093e-17]
[ -0.707106781187 3.12521276219e-17 0.353553390593
6.96057794736e-17 -0.353553390593 -1.72861506093e-17]
[ -0.0 0.0 -0.0
0.0 0.0 -0.5]
[ -0.0 0.816496580928 5.51587855252e-17
-0.288675134595 -7.45998857306e-17 1.68969994439e-17]
[ -0.0 0.0 -0.707106781187
9.67672224796e-18 5.13672629661e-18 0.0]
[ -0.0 0.0 -0.0
0.0 0.0 0.5]
[ -0.0 0.0 -0.0
0.866025403784 -2.24700900248e-17 -1.5111067779e-17]
[ -0.0 0.0 -0.0
0.0 0.707106781187 1.24852656425e-17]
[ -0.0 0.0 -0.0
0.0 0.0 0.5]
```

Does anybody know where I've gone wrong here? Is there another more reliable method for computing an orthonormal basis that I could use? Thanks again for putting up with a newbie!

best regards

Brian