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I have a set of four vectors in $\mathbb{R}^4$: $\{ \vec v_1, \vec v_2, \vec v_3, \vec v_4 \}$ The first three are linearly independent, but $ \vec v_4 $ is a linear combination of the others. Is it possible to find a vector in $\mathbb{R}^4$ that is orthogonal to $\{ \vec v_1, \vec v_2, \vec v_3, \vec v_4 \}$ ?

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Yes. Find a vector orthogonal to $\vec v_1, \vec v_2, \vec v_3$ (by Gram-Schmidt). It is automatically also orthogonal to $\vec v_4$.

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