Let V be the three dimensional subspace of $$R^{4}$$ defined by the equation $$x_{1} − 2x_{2} + 3x_{3} − 4x_{4} = 0.$$ Find an orthonormal basis for V .
I know that I should use Gram-Schmidt but I am unsure of how to apply it. I saw the examples and other problems dealing with multiple basis here but it did not make sense to me.
Please explain it to me in detail so that I can understand.
So I found a basis to be $$u_{1}=(\frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}},0,0)\\ u_{2}=(0,0,0,0)\\u_3=(\frac{4}{5\sqrt{\frac{41}{25}}},0,0,\frac{1}{\sqrt\frac{41}{25}})$$
As a first step find a "simple" basis that is
and, we are lucky, $v_1$ and $v_2$ are othogonal, then we need to orthogonalize $v_3$ with respect to $v_1$ and $v_2$ and finally normalize the three orthogonal vectors.
Refer also to the related
Notably we can find the orthogonal component of $v_3$ by
$$w_3=v_3-\frac{v_1^Tw_3}{v_1^Tv_1}v_1-\frac{v_2^Tw_3}{v_2^Tv_2}v_2$$
and then normalize by
$$u_1=\frac{v_1}{\sqrt{v_1^Tv_1}}\quad u_2=\frac{v_2}{\sqrt{v_2^Tv_2}}\quad u_3=\frac{w_3}{\sqrt{w_3^Tw_3}}$$