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

In summary, there is a proof that if the vector v1 is a unit vector in R^n, there exists an orthogonal matrix A with v1 as its first column. This can be achieved by using the Gram-Schmidt orthogonalization process on a basis for R containing v1 and normalizing the resulting orthogonal basis, or by using any rotation matrix that rotates the vector [1,0,0,...] into v1. There are no major assumptions overlooked in the proof.
  • #1
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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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  • #2
Nothing- although I would say "Let [itex]\beta[/itex] be a basis for R containing v1..."
 
  • #3
Another candidate is any rotation matrix which rotates the vector [1,0,0,...] into v_1.
 
  • #4
HallsofIvy said:
Nothing- although I would say "Let [itex]\beta[/itex] 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.
 

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

1. How do you form an orthogonal matrix?

An orthogonal matrix is formed by taking a set of linearly independent unit vectors and arranging them as the columns of a matrix. This means that the columns of the matrix are perpendicular to each other, and each column has a magnitude of 1.

2. What is the significance of the first column being a unit vector in forming an orthogonal matrix?

In order for a matrix to be orthogonal, the first column must be a unit vector. This ensures that the first column is perpendicular to all other columns, satisfying the definition of an orthogonal matrix.

3. How do you ensure that the first column of an orthogonal matrix is a given unit vector?

To ensure that the first column of an orthogonal matrix is a given unit vector, simply divide the vector by its magnitude. This will result in a unit vector with the same direction as the original vector.

4. Can an orthogonal matrix have more than one unit vector as its first column?

No, by definition, an orthogonal matrix can only have one unit vector as its first column. If there were multiple unit vectors, the columns would not all be perpendicular to each other.

5. Why is forming an orthogonal matrix with a given 1st column important?

An orthogonal matrix with a given first column is important because it allows for a specific orientation and magnitude of the first column, while still satisfying the properties of an orthogonal matrix. This can be useful in applications such as rotation transformations in linear algebra.

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