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)}:
$\displaystyle (1,0,0) \cdot (1,1,0) = 1$
{(1,0,0), (1,-1,0), (0,0,1)}:
$\displaystyle (1,0,0) \cdot (1,-1,0) = 1$
{(1,0,1), (1,0,-1), (0,1,0)}:
This one is orthogonal:
$\displaystyle (1,0,1) \cdot (1,0,-1) = 0$
$\displaystyle (1,0,1) \cdot (0,1,0) = 0$
$\displaystyle (1,0,-1) \cdot (0,1,0) = 0$
To tell if the vectors are independent, set $\displaystyle r_1(1,0,1)+r_2(1,0,-1)+r_3(0,1,0)=(0,0,0)$. Then:
$\displaystyle r_1+r_2=0$
$\displaystyle r_3=0$
$\displaystyle r_1-r_2=0$
and you can see the only solution is $\displaystyle r_1=r_2=r_3=0$, which means the vectors are linearly independent.
{(1,0,0), (1,1,0), (1,1,1)}:
$\displaystyle (1,0,0) \cdot (1,1,0) = 1$
So the third set is an orthogonal basis, and the others aren't.
- Hollywood