Could anyone tell me if any of the following sets of vectors is an orthogonal basis for R^{3}
{(1,0,0), (1,1,0), (0,0,1)}
{(1,0,0), (1,-1,0), (0,0,1)}
{(1,0,1), (1,0,-1), (0,1,0)}
{(1,0,0), (1,1,0), (1,1,1)}
It's easy to eliminate three of the options by checking if they're orthogonal:
{(1,0,0), (1,1,0), (0,0,1)}:
{(1,0,0), (1,-1,0), (0,0,1)}:
{(1,0,1), (1,0,-1), (0,1,0)}:
This one is orthogonal:
To tell if the vectors are independent, set . Then:
and you can see the only solution is , which means the vectors are linearly independent.
{(1,0,0), (1,1,0), (1,1,1)}:
So the third set is an orthogonal basis, and the others aren't.
- Hollywood