Are These Subsets of R3 Subspaces?

[MoreDrinks]

Homework Statement

Which of the following subsets of R³ are subspaces? The set of all vectors of the form (a,b,c) where a, b, and c are...

Homework Equations

1. integers
2. rational numbers

The Attempt at a Solution

I think neither are subspaces. IIRC, the scalar just needs to be from R³ and not, for example, an integer for 1 or a rational number for 2.

So for number 1, I can multiply the integers of vector (a,b,c) by some non-integer k, ending up with (ka,kb,kc) outside the subset, and thus not a subspace.

For number 2, I can multiply the rational numbers of vector (a,b,c) some some irrational number (say, ∏) and end up with (∏a, ∏b, ∏c), all outside the subset and thus not a subspace.

Or am I totally wrong?

 

[jbunniii]

Or am I totally wrong?

No, you are totally correct. The indicated sets are not subspaces of $\mathbb{R}^3$, for the reasons you stated.

 

[jbunniii]

the scalar just needs to be from R³

Correction: the scalars are elements of $\mathbb{R}$, not $\mathbb{R}^3$.

 

[MoreDrinks]

Correction: the scalars are elements of $\mathbb{R}$, not $\mathbb{R}^3$.

If we're dealing with complex space, can scalars be complex?

Thanks for the help!

 

[jbunniii]

If we're dealing with complex space, can scalars be complex?

Thanks for the help!

They can, but then it wouldn't be $\mathbb{R}^3$ anymore. It would be $\mathbb{C}^3$.

 

[MoreDrinks]

They can, but then it wouldn't be $\mathbb{R}^3$ anymore. It would be $\mathbb{C}^3$.

True, thanks! Would the correct term be that we're working in the "field" of R^3 or just R^3 space when talking about this?

 

[jbunniii]

True, thanks! Would the correct term be that we're working in the "field" of R^3 or just R^3 space when talking about this?

To be precise, a vector space consists of an abelian group of vectors and a field of scalars, along with some rules governing the multiplication of a vector by a scalar.

So if we want to be precise, we would say that we are working in the vector space in which the vectors are elements of $\mathbb{R}^3$ and the scalars are elements of $\mathbb{R}$, with the usual rules of multiplication.

However, for brevity we typically say that we are working in the vector space $\mathbb{R}^3$, and unless stated otherwise, it is understood that the scalar field is $\mathbb{R}$.

Similarly, we may say that we are working in the vector space $\mathbb{C}^3$, where the assumption is that unless stated otherwise, the scalar field is $\mathbb{C}$.

 

[MoreDrinks]

To be precise, a vector space consists of an abelian group of vectors and a field of scalars, along with some rules governing the multiplication of a vector by a scalar.

So if we want to be precise, we would say that we are working in the vector space in which the vectors are elements of $\mathbb{R}^3$ and the scalars are elements of $\mathbb{R}$, with the usual rules of multiplication.

However, for brevity we typically say that we are working in the vector space $\mathbb{R}^3$, and unless stated otherwise, it is understood that the scalar field is $\mathbb{R}$.

Similarly, we may say that we are working in the vector space $\mathbb{C}^3$, where the assumption is that unless stated otherwise, the scalar field is $\mathbb{C}$.

Thank you, that clears up a lot.