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## Homework Statement

Show that if the vector [tex]\textbf{v}_1[/tex] is a unit vector (presumably in [tex]\Re^n[/tex]) then we can find an orthogonal matrix [tex]\textit{A}[/tex] that has as its first column the vector [tex]\textbf{v}_1[/tex].

## The Attempt at a Solution

This seems to be trivially easy. Suppose we have a basis [tex]\beta[/tex] for [tex]\Re^n[/tex]. We may apply the Gram-Schmidt orthogonalization process to [tex]\beta[/tex] with [tex]\textbf{v}_1[/tex] as the generating vector and normalise the resultant orthogonal basis to obtain an orthonormal basis [tex]\gamma[/tex]. Choose [tex]\textit{A}[/tex] such that the elements of [tex]\gamma[/tex] comprise the columns of [tex]\textit{A}[/tex].

QED

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So what am I overlooking?

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