Question

Does the basis 

B = {(1, 0, 1), (1, 0, — 1), (0, 3, 4)} 

form an orthonormal basis of R3 with 

respect to the standard inner product of 

R3 ? Justify your answer. If it doesn't form 

an orthonormal basis for R3, apply 

Gram-Schmidt process to obtain an 

orthonormal basis R3 with respect to the 

standard inner product on R3. 

Answer 1

They do not form an orthonormal basis, as none of this vectors is of a norm 1 and in addition $(v_1, v_3) \neq 0$. Let's apply the Gram-Schmidt procedure:

1. $e_1 = \frac{v_1}{||v_1||} = (\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})$
2. $e_2 = \frac{v_2 - (v_2, e_1) e_1}{|| v_2 - (v_2, e_1)e_1||} =\frac{v_2}{||v_2||} = (\frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}})$
3. $e_3 = \frac{v_3 - (v_3,e_1)e_1 - (v_3,e_2) e_2}{||v_3 - (v_3,e_1)e_1 - (v_3,e_2) e_2||}= \frac{(0,3,4) - (2,0,2)-(-2,0,2)}{||(0,3,4) - (2,0,2)-(-2,0,2)||}=(0,1,0)$