How to quickly check if vectors are an orthonormal basis of a vector space?
According to the above link, we can check if a set of vectors are an orthonormal basis of some vector space $V$ by checking if the vectors in the set are:
- all orthogonal to each other: "ortho"
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all of unit length: "normal"
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(if the set of vectors are orthonormal, then the set of vectors are linearly independent)
But doesn't this just show that we have an orthonormal set of vectors?
How do we know the set of vectors are an orthonormal basis of some vector space $V$? In other words how do we know the set of vectors span the vector space $V$?
Best Answer
The set of vectors should also span the whole space.
If we know the dimension of the space then we should simply count the orthonormal vectors and make sure that we have as many vectors in our basis as the dimension of the space.
Otherwise, we have to prove that every vector is a linear combination of the basis vectors.