Forming an orthogonal matrix whose 1st column is a given unit vector

[v0id]

Homework Statement

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

The Attempt at a Solution

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

QED

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So what am I overlooking?

 

[HallsofIvy]

Nothing- although I would say "Let $\beta$ be a basis for R containing v₁..."

 

[StatusX]

Another candidate is any rotation matrix which rotates the vector [1,0,0,...] into v_1.

 

[v0id]

Nothing- although I would say "Let $\beta$ be a basis for R containing v₁..."

Thanks. I thought I had made a huge assumption somewhere.

StatusX, yes that's a simpler possibility indeed.