Is it Possible to Have Only Two Subspaces in a Vector Space?

[gajohnson]

Homework Statement

When is it true that the only subspaces of a vector space V, are V and {0}?

Homework Equations

NA

The Attempt at a Solution

Because a subspace has to be closed under addition and scalar multiplication, it is my intuition that this is true only when there are no infinite subsets of V. However, I am not sure this is correct and I do not have a better attempt at an answer. Any help is greatly appreciated.

 

[LCKurtz]

What about the real line?

 

[Michael Redei]

Infinity doesn't enter the matter, since there are finite vector spaces with subspaces "in between" {0} and themselves.

Take a vector space V and some subspace S of V, where S is neither just {0} nor all of V.

What does S≠{0} mean, what must S look like so it's not just {0}?

And what must V be like, so that this subspace S automatically becomes equal to V if it's not just {0}?

 

[LCKurtz]

Infinity doesn't enter the matter, since there are finite vector spaces with subspaces "in between" {0} and themselves.

Take a vector space V and some subspace S of V, where S is neither just {0} nor all of V.

What does S≠{0} mean, what must S look like so it's not just {0}?

And what must V be like, so that this subspace S automatically becomes equal to V if it's not just {0}?

By a "finite vector space" do you mean just {0}?

 

[gajohnson]

Infinity doesn't enter the matter, since there are finite vector spaces with subspaces "in between" {0} and themselves.

Take a vector space V and some subspace S of V, where S is neither just {0} nor all of V.

What does S≠{0} mean, what must S look like so it's not just {0}?

And what must V be like, so that this subspace S automatically becomes equal to V if it's not just {0}?

S would have to have dimension greater than 0 and less than dimV. In which case it seems like it would be a subspace. Does this mean it's true that the only subspaces of V are V and {0} only when V is {0}? I'm probably way off track here...

 

[Michael Redei]

By a "finite vector space" do you mean just {0}?

No, I'm thinking of vector spaces over finite fields.

S would have to have dimension greater than 0 and less than dimV. In which case it seems like it would be a subspace. Does this mean it's true that the only subspaces of V are V and {0} only when V is {0}? I'm probably way off track here...

Have a look at some examples: Look at V = {0}, V = a line, V = a plane, V = $\mathbb R^3$. In which of these cases are there no subspaces "in between" {0} and V? If you see any pattern there, how about $\mathbb R^4$ etc.?

 

[gajohnson]

No, I'm thinking of vector spaces over finite fields.



Have a look at some examples: Look at V = {0}, V = a line, V = a plane, V = $\mathbb R^3$. In which of these cases are there no subspaces "in between" {0} and V? If you see any pattern there, how about $\mathbb R^4$ etc.?

It seems like it should be true only when dimV<2. If dimV is 1, then a subspace of V could only have dim1 or 0, thus making it true that the only subspaces are V itself and {0}. If dimV≥2, then V can have at least have a subspace with dim1. Is this right?

 

[Dick]

It seems like it should be true only when dimV<2. If dimV is 1, then a subspace of V could only have dim1 or 0, thus making it true that the only subspaces are V itself and {0}. If dimV≥2, then V can have at least have a subspace with dim1. Is this right?

Yes, it is. Dimension is the number of vectors in a basis. If dim V>=2 then pick anyone of the vectors and its span is a subspace of dimension 1.

 

[gajohnson]

Yes, it is. Dimension is the number of vectors in a basis. If dim V>=2 then pick anyone of the vectors and its span is a subspace of dimension 1.

Great, thanks!