General Question on Subspaces

  • #1
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
 

Answers and Replies

  • #2
22,089
3,290
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.
 
  • #3
mathman
Science Advisor
7,877
453
As a practical matter {0} is zero dimensional. Most discussions about subspaces usually assume at least one dimension.
 

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