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A linear algebra homework question has me stumped.

Let $S$ be a subspace of $\mathbb{R}^4$ which contains all vectors $[x_1, x_2, x_3, x_4]^T$ for which the following must be true: $x_1 + x_2 + x_3 + x_4 = 0$

Find a basis for the subspace $S^\perp$ where all vectors are orthogonal on $S$.

I'm having trouble understanding how to proceed with this. I first tried to write 4 vectors and use the Gram-Schmidt to construct this basis, but I can't seem to find 4 linear independent vectors. This was the only solution from my perspective but I can't seem to figure it out.

Thanks

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  • $\begingroup$ Well, there definitely won't be $4$ linearly independent vectors; $S$ only has dimension $3$! (If you are new to linear algebra then this won't be obvious, but it is important to know there's no reason to think $S$ should be 4-dimensional in advance!) $\endgroup$ – preferred_anon Mar 10 '17 at 17:42
  • $\begingroup$ The wording is somewhat unclear. In fact, as written it seems to ask for a basis of $S^{\top}$---which is only $1$-dimensional---not $S$. $\endgroup$ – Travis Willse Mar 10 '17 at 20:04
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Hint We can write the equation $x_1 + x_2 + x_3 + x_4 = 0$ defining $S$ as $$[x_1, x_2, x_3, x_4]^{\top} \cdot [1, 1, 1, 1]^{\top} = 0.$$

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