Why is the dimension of the vector space , 0?

[Dosmascerveza]

In other words, why is dim[{0}]=0. My math professor explained that since the 0 vector is just a POINT in R2 that the zero subspace doesn't have a basis and therefore has dimension zero. This is not satisfactory.

For example, I know R2 has a dimension 2, P_n has dimension n+1, M_(2,2) has dimension 4. These numbers make sense to me but if the zero subspace contain the zero vector why does it not have a basis?

Can someone explain this concept to me a bit further

 

[Werg22]

Convention. It makes, among other things, the rank-nullity theorem work.

 

[Dosmascerveza]

So you say dim[{o}]=0 because you've defined it that way. So the definition of dimension is different for the zero subspace than for every other V space? But how can there be two definitions of dimension be used in the same sense?

 

[Hurkyl]

the zero subspace doesn't have a basis and therefore has dimension zero.

That statement is false; are you sure you heard him correctly? The zero subspace does have a basis -- the empty set.

 

[Dosmascerveza]

I heard him correctly. He made no mention of the empty set. He may have taken some "liberties" to keep it simple for the country-folk.

 

[disregardthat]

That statement is false; are you sure you heard him correctly? The zero subspace does have a basis -- the empty set.

Isn't the basis supposed to span the vector space? The empty set does not even span the the null-vector.
In any case, {0} can hardly be treated as a basis, because it is not linearly independent! It is common however to treat trivial cases with "arbitrary" definitions to make general rules hold for these cases as well. Compare with the convention 0^0 = 0 in power series.

 

[Hurkyl]

Isn't the basis supposed to span the vector space? The empty set does not even span the the null-vector.

The linear combination of no terms is equal to zero.

I heard him correctly. He made no mention of the empty set. He may have taken some "liberties" to keep it simple for the country-folk.

I guess I don't find that too surprising -- many people seem to vehemently detest dealing with degenerate cases.

 

[Dosmascerveza]

Well thanks Hurkyl! I will approach him with your assertion and we will go from there. I hope pressing for some deeper answers won't adversely affect my grade.