Show aβ is a Basis for ℝ over Q

[SMA_01]

Let β be a basis for ℝ over Q (the set of all rational numbers) and let a\inℝ, a≠1.

Show that aβ={ay|y\inβ} is a basis for ℝ over Q for all a≠0.

So I need to show (1) Linear independence, and (2) spanning. I am a little confused, especially because the dimension for the vector space is uncountably infinite.
Here is what I have for linear independence:

I took a finite subset of aβ, {az|z\inS\subsetβ, |S|<|ℝ|} (not sure if I wrote that correctly), so if

\sum_{z\in S}az=0

then z must be zero since a≠0 (initial assumption). Therefore, it follows that aβ is linearly independent.

Is this correct?
I don't know how to approach spanning.

Thanks in advance.

 

[Mark44]

Let β be a basis for ℝ over Q (the set of all rational numbers) and let a\inℝ, a≠1.

Show that aβ={ay|y\inβ} is a basis for ℝ over Q for all a≠0.

So I need to show (1) Linear independence, and (2) spanning. I am a little confused, especially because the dimension for the vector space is uncountably infinite.
Here is what I have for linear independence:

I took a finite subset of aβ, {az|z\inS\subsetβ, |S|<|ℝ|} (not sure if I wrote that correctly), so if

\sum_{z\in S}az=0

then z must be zero since a≠0 (initial assumption).

This seems like armwaving to me.

Therefore, it follows that aβ is linearly independent.

Is this correct?
I don't know how to approach spanning.

Do you have a sense of what this problem is about? In particular, what a basis over Q that spans R might look like? For example, can you list a few of the basis elements of such a basis?

Given that β is some basis, you need to show that aβ is also a basis, where a ≠ 0.

To show that any set is a basis for some space, you need to do two things:
1. Show that the set of basis elements is linearly independent.
2. Show that the set of basis elements spans the space.

The latter requirement means that if x $\in$ R, then x can be written as a linear combination of the basis elements.

 

[SMA_01]

Do you have a sense of what this problem is about? In particular, what a basis over Q that spans R might look like? For example, can you list a few of the basis elements of such a basis?

Honestly, I can't fully grasp it intuitively. I know how a basis is for ℝ with n-dimension, but not ℝ.
I just can't see how something can span ℝ, is the standard basis applicable here?

Given that β is some basis, you need to show that aβ is also a basis, where a ≠ 0.

To show that any set is a basis for some space, you need to do two things:
1. Show that the set of basis elements is linearly independent.
2. Show that the set of basis elements spans the space.

The latter requirement means that if x $\in$ R, then x can be written as a linear combination of the basis elements.

If I can't show linear independence/spanning the usual way, how should I do this?


Thanks.

 

[vela]

You do show independence the usual way. You need to show that for any finite subset $\{v_1,v_2,\dots,v_n\} \subset a\beta$ that
$$q_1 v_1 + q_2 v_2 + \cdots + q_n v_n = 0$$ implies $q_1=q_2=\cdots=q_n=0$, where $q_i \in \mathbb{Q}$. What you wrote above doesn't show this. For one thing, there's no mention of the $q_i$'s.

 

[SMA_01]

You do show independence the usual way. You need to show that for any finite subset $\{v_1,v_2,\dots,v_n\} \subset a\beta$ that
$$q_1 v_1 + q_2 v_2 + \cdots + q_n v_n = 0$$ implies $q_1=q_2=\cdots=q_n=0$, where $q_i \in \mathbb{Q}$. What you wrote above doesn't show this. For one thing, there's no mention of the $q_i$'s.

But a basis for ℝ over Q is uncountable, so how can I take a finite, countable subset?

 

[Dick]

But a basis for ℝ over Q is uncountable, so how can I take a finite, countable subset?

Because that's the definition of linear independence. To show it's linearly independent you need to show every finite subset is linearly independent.

 

[SMA_01]

Because that's the definition of linear independence. To show it's linearly independent you need to show every finite subset is linearly independent.

Okay, correct me if I'm wrong, but if I take a finite subset, I thought it won't be countable? Like I can't order the terms...

 

[vela]

Finite implies countable.

 

[SMA_01]

Finite implies countable.

So if I take a finite subset of ℝ, then I can order the terms? Sorry, I just want to clarify.

 

[vela]

What's the precise definition of finite? If you know this basic definition, you should be able to answer your own question.