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I need to find the orthogonal complement in $\mathbb{R}^4$ of the space spanned by $(1,0,1,1)$ and $(-1,2,0,0)$. Here is what I am thinking: Let's call $A$ the space spanned by $(1,0,1,1)$ and $(-1,2,0,0)$. A vector $v=(v_1,v_2,v_3,v_4)$ that exists in the orthogonal complement $A^\perp$ should satisfy

$$(v_1,v_2,v_3,v_4)\cdot (1,0,1,1) =0$$ and $$ (v_1,v_2,v_3,v_4)\cdot (-1,2,0,0)=0$$ This means that $v_1+v_3+v_4=0$ and $-v_1+2v_2=0.$

Can someone give me a hint as to whether this thinking is correct? Should I now just try to find a basis for this space?

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Your approach is correct. Note that $$dim \Bbb R^4=4=dim A+dim A\perp$$ Therefore dim$A^\perp=2$ Solving the linear system we find that:$$v_1+v_3+v_4=0, v_1+2v_2=0$$ Therefore a generic vector of $A^\perp$ is equal : $v=(-v_3+v_4,-\frac {1}{2}(v_3+v_4),v_3,v_4)$. $A^\perp$ is spanned by $(-1,-\frac{1}{2},1,0)$ and $(-1,1,0,1)$

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Hint: $v_1=-v_3-v_4$ and $v_2=-\frac12(v_3+v_4)$.

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