Can a Vector Space Over Field F Contain Entries from Other Fields?

[zcd]

Just for clarification, if a vector space is defined over a field F, are entries inside the vectors in the vector space necessarily restricted to field F? Say I had a vector space V={(a₁,a₂,...):a_i∈C} , could the vector space be over the field R so that I only take scalars from the reals?

 

[mathwonk]

well,yes, but what is really your question?

 

[zcd]

My question is do entries in an n-tuple from a vector space have to be from the same field as the field the vector space is defined over?

 

[micromass]

No, they don't. And elements of a vector space don't need to be tuple. You only need to be able to multiplicate them with elements of F. But is it not necessary that elements of the vector space have anything to do with F.

However, every (finite dimensional space) is isomorphic to Fⁿ. So in fact we are working with tuples anyway, even if we don't realize it...

 

[Fredrik]

A good example is the set of complex traceless self-adjoint 2×2 matrices. The standard definitions of addition and scalar multiplication give it the structure of a vector space over ℝ. But the set isn't closed under multiplication by i, so the same standard definitions do not give it the structure of a vector space over ℂ.

It's easy to show that this vector space is 3-dimensional, and that the set of Pauli spin matrices is a basis. With an appropriate choice of inner product, this basis is orthonormal.

 

[mathwonk]

I'm sorry, this question is so imprecise as not to make sense to me, but the others seem to know what you mean.

Are you asking whether, say the complex numbers can be considered as a vector space over the real numbers? Then the answer is yes, but in writing them down as "tuples" one would then use real numbers to represent them, not complex numbers. So the phrase "entries inside the vectors" is the part that is too imprecise for me to understand perfectly.

I.e. when you say this, I implicitly assume you mean entries in a basis representation, which do come from the base field, but that is apparently not what you meant.

 

[Fredrik]

So the phrase "entries inside the vectors" is the part that is too imprecise for me to understand perfectly.

When he says "vector" he means "tuple", not "member of a vector space". So he's asking if ℂⁿ can be given the structure of a vector space over ℝ.

I.e. when you say this, I implicitly assume you mean entries in a basis representation,

I'm pretty sure he means entries in the standard basis, not any other.

Then the answer is yes, but in writing them down as "tuples" one would then use real numbers to represent them, not complex numbers.

Not necessarily. I don't see why anyone would want to do this, but you can give ℂⁿ the structure of a vector space over ℝ, by first defining the standard (complex) vector space structure and then restricting the scalar multiplication function to ℝ×ℂⁿ.

The example I used in my previous post is much more interesting. It's a 3-dimensional vector space over ℝ, and the basis I mentioned looks like this:

$$\sigma_1=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\quad \sigma_2=\begin{pmatrix}0 & -i\\ i & 0\end{pmatrix},\quad \sigma_3=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$$

So an arbitrary member can be written as

$$x=\sum_{k=1}^3 x_k\sigma_k=\begin{pmatrix}x_3 & x_1-ix_2\\ x_1+ix_2 & -x_3\end{pmatrix}$$

Now, the tuple that you and I would associate with this vector and this basis is $(x_1,x_2,x_3)$, but the OP's phrase "inside the vector" refers to what's inside the matrix on the right-hand side.

 

[zcd]

I'm sorry, this question is so imprecise as not to make sense to me, but the others seem to know what you mean.

Sorry for not being clear with my question. I'm asking whether the field a particular vector space is over has any bearing on what's inside a vector from the vector space, or are those two separate matters?

The example I used in my previous post is much more interesting. It's a 3-dimensional vector space over ℝ, and the basis I mentioned looks like this:

$$\sigma_1=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\quad \sigma_2=\begin{pmatrix}0 & -i\\ i & 0\end{pmatrix},\quad \sigma_3=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$$

So an arbitrary member can be written as

$$x=\sum_{k=1}^3 x_k\sigma_k=\begin{pmatrix}x_3 & x_1-ix_2\\ x_1+ix_2 & -x_3\end{pmatrix}$$

Now, the tuple that you and I would associate with this vector and this basis is $(x_1,x_2,x_3)$, but the OP's phrase "inside the vector" refers to what's inside the matrix on the right-hand side.

So what you're doing is using a vector space of 2x2 matrices with complex entries, but since the scalars used for scalar multiplication are all reals the vector space is over the field of reals?

 

[Fredrik]

So what you're doing is using a vector space of 2x2 matrices with complex entries, but since the scalars used for scalar multiplication are all reals the vector space is over the field of reals?

Exactly. One of the things that makes this example interesting is that we don't have the option to choose the field of scalars to be ℂ, because if A is a member of the set, iA isn't. (iA)*=-iA*=-iA≠iA.

Sorry for not being clear with my question. I'm asking whether the field a particular vector space is over has any bearing on what's inside a vector from the vector space, or are those two separate matters?

It's the "inside a vector" part of your statement that's unclear. "Vector" means "member of a vector space", not "member of Fⁿ, where F is some field", so it's not at all clear what "inside a vector" means. A natural interpretation would be that it's a reference to the members of the matrix of components of the vector in a given basis. In my example, that would be the real triple (x₁,x₂,x₃), not the complex 4-tuple x.

 

[Landau]

I think the only sensible answer to this last question is "yes, the two are separate matters". A vector space over the field K is nothing but an abelian additive group with a action of K on it. The members of that group do not have to have any "connection" with K (whatever that means), only a sensible action of K on it must be defined so that the product of an element of K and an element of the group yields an element of the group.

 

[zcd]

Thanks for the clarification!