Do All Subspaces in Rn Have Orthonormal Bases?

[fredrogers3]

Hello everyone,
I have a theoretical question on subspaces. Consider the space Rⁿ. The zero vector is indeed a subspace of Rⁿ. However, if I am not mistaken, the zero vector has no orthonormal basis, even though it is a subspace. I thought all subspaces have an orthonormal basis (or is it all subspaces of n dimensions?). Can anyone shed any light on where I am going wrong here?

Thanks

 

[micromass]

Hello everyone,
I have a theoretical question on subspaces. Consider the space Rⁿ. The zero vector is indeed a subspace of Rⁿ.

Well, technically speaking, the subspace is not the zero vector but the set containing the zero vector. That is $\{0\}$ is the subspace of $\mathbb{R}^n$.

However, if I am not mistaken, the zero vector has no orthonormal basis, even though it is a subspace. I thought all subspaces have an orthonormal basis (or is it all subspaces of n dimensions?). Can anyone shed any light on where I am going wrong here?

The empty set would be the orthonormal basis for $\{0\}$. Depending on your definitions, this is either a convention or provable.

For a subset $X$ of a vector space $V$, I can define $\textrm{span}(X)$ the smallest subspace of $V$ containing $X$. Under this definition, we see that $\textrm{span}(\emptyset) = \{0\}$. Furthermore, it is vacuous truth that any two elements in $\emptyset$ are orthogonal since it has no elements to begin with.

 

[mathman]

As a practical matter {0} is zero dimensional. Most discussions about subspaces usually assume at least one dimension.