Chapter 6 - Orthogonality and Least Squares - 6.1 Exercises - Page 339: 29

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Each vector in W can be written as a linear combination of the vectors $\vec{v_1}$ through $\vec{v_p}$. Because x is orthogonal to each of these vectors, it is orthogonal to every possible linear combination of these vectors, and is thus orthogonal to each v in W.