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

1. Nov 30, 2006

### v0id

1. The problem statement, all variables and given/known data
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 $$\textbf{v}_1$$.

3. 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?

Last edited: Nov 30, 2006
2. Dec 1, 2006

### HallsofIvy

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

3. Dec 1, 2006

### StatusX

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

4. Dec 1, 2006