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I have to find an orthonormal basis for $ \Bbb R^4 $ which includes the vectors $u_1 = (\frac{1}{\sqrt2}, 0,\frac{1}{\sqrt2}, 0)$ and $u_2 = (\frac{-1}2, \frac1 2,\frac1 2, \frac{-1}2)$

I really do not know how to do it, I tried to use the Gram-Schmidt method but when I started with $u_1$ then obtained $u_2$.

Any ideas?

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  • $\begingroup$ First, find two vectors $u_3$ and $u_4$ which are perpendicular to $u_1$ and $u_2$. Then do Gram-Schmidt on $u_3$ and $u_4$ (to get $u_3$ and $v_4$). Now you will have an orthogonal basis. Then normalize $u_3$ and $v_4$ (to get $w_3$ and $w_4$); then $\{ u_1,u_2,w_3, w_4\}$ will be an orthonormal basis. $\endgroup$ – Christopher Carl Heckman May 9 '16 at 0:28
  • $\begingroup$ How can I find the two vectors u3 and u4 which are perpendicular to $u_1$ and $u_2$. $\endgroup$ – Alex Turner May 9 '16 at 0:34
  • $\begingroup$ Set up a system of linear equations, solve it, and then parameterize the solutions. If you write your solution in vector form $x = ru + sv$, then $u$ and $v$ will be perpendicular to $u_1$ and $u_2$ (and be linearly independent). $\endgroup$ – Christopher Carl Heckman May 9 '16 at 0:35
  • $\begingroup$ You really only need two vectors that together with $u_1$ and $u_2$ form a linearly independent set. G-S will crank out perpendicular vectors. You could choose $(1,0,0,0)$ and $(0,1,0,0)$, for instance. $\endgroup$ – amd May 9 '16 at 7:35
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Aside from the normalization constants, the vectors can be $\left(1,0,1,0\right)$, $\left(-1,1,1,-1\right)$, $\left(0,1,0,1\right)$, $\left(-1,-1,1,1\right)$

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