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Let $\mathcal{S} \subset \mathbb{R}^{4}$ be the vectors $X=\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ that satisfy $x_{1}+x_{2}-x_{3}+x_{4}=0$.
a) What is the dimension of $\mathcal{S}$ ?
b) Find a basis for the orthogonal complement of $\mathcal{S}$.
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