I have a question which says the following:
Let $V$ be the span of $v_{1}=(0,1,2)$, $v_{2}=(-1,0,1)$ and $v_{3}=(-1,1,3)$.
Construct an orthonormal basis $B'$ for $V$ (usual dot product).
I know how to do the question. That's easy. I need to apply the gram schmidt procedure thrice and I have my answer. My problem is the 3rd iteration.
$U_{1}=V_{1}=(0,1,2)$
Orthonormalizing, we obtain $V_{1}'=(0,\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}})$
Then $U_2=V_2-\frac{V_2 \cdot U_1}{||U_1||^2}U_1$
Doing so gives: $U_2=(-1,\frac{-2}{5},\frac{1}{5})$
Orthonormalizing, we get $V_{2}'=(0,\frac{-2}{\sqrt{30}},\frac{1}{\sqrt{30}})$
Finally, $U_3=V_3-\frac{V_3 \cdot U_1}{||U_1||^2}U_1 - \frac{V_3 \cdot U_2}{||U_2||^2}U_2$
However this time, if I plug in everything, I get $U_3=(0,0,0)$. While that is indeed orthogonal to both ${v_1}$ and ${v_2}$ there is no way I can orthonormalize $U_3$ since I get division by $0$.
Is the question just badly asked? Because there is simply no way to obtain an orthonormal basis.
Hint: what is the dimension of $V$? This should tell you how big a basis should be.