Is Q a vector space over Z or Q?
$begingroup$
Is $Q^n$ a vector space over $Z$ or over $Q$?
$Q^n$ is clearly not a vector space over $R$, because scalar multiplication of some $q in Q$ by $pi$ renders $pi q$, which is irrational.
linear-algebra vector-spaces vectors
asked 42 mins ago
CactusCactus
211
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Cactus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$
add a comment |
$begingroup$
Is $Q^n$ a vector space over $Z$ or over $Q$?
$Q^n$ is clearly not a vector space over $R$, because scalar multiplication of some $q in Q$ by $pi$ renders $pi q$, which is irrational.
linear-algebra vector-spaces vectors
asked 42 mins ago
CactusCactus
211
New contributor
Cactus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
4
$begingroup$
It doesn't make sense to be a vector space over $mathbb Z$, because $mathbb Z$ is not a field. However, it is a vector space over $mathbb Q$.
$endgroup$
– Ashwin Trisal
41 mins ago
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
39 mins ago
add a comment |
$begingroup$
Is $Q^n$ a vector space over $Z$ or over $Q$?
$Q^n$ is clearly not a vector space over $R$, because scalar multiplication of some $q in Q$ by $pi$ renders $pi q$, which is irrational.
linear-algebra vector-spaces vectors
asked 42 mins ago
CactusCactus
211
New contributor
Cactus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Is $Q^n$ a vector space over $Z$ or over $Q$?
$Q^n$ is clearly not a vector space over $R$, because scalar multiplication of some $q in Q$ by $pi$ renders $pi q$, which is irrational.
linear-algebra vector-spaces vectors
linear-algebra vector-spaces vectors
asked 42 mins ago
CactusCactus
211
New contributor
Cactus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 42 mins ago
CactusCactus
211
New contributor
Cactus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 42 mins ago
CactusCactus
211
New contributor
Cactus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 42 mins ago
CactusCactus
211
asked 42 mins ago
CactusCactus
211
211
New contributor
Cactus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Cactus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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4
$begingroup$
It doesn't make sense to be a vector space over $mathbb Z$, because $mathbb Z$ is not a field. However, it is a vector space over $mathbb Q$.
$endgroup$
– Ashwin Trisal
41 mins ago
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
39 mins ago
add a comment |
4
$begingroup$
It doesn't make sense to be a vector space over $mathbb Z$, because $mathbb Z$ is not a field. However, it is a vector space over $mathbb Q$.
$endgroup$
– Ashwin Trisal
41 mins ago
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
39 mins ago
4
4
$begingroup$
It doesn't make sense to be a vector space over $mathbb Z$, because $mathbb Z$ is not a field. However, it is a vector space over $mathbb Q$.
$endgroup$
– Ashwin Trisal
41 mins ago
$begingroup$
It doesn't make sense to be a vector space over $mathbb Z$, because $mathbb Z$ is not a field. However, it is a vector space over $mathbb Q$.
$endgroup$
– Ashwin Trisal
41 mins ago
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
39 mins ago
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
39 mins ago
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
You can't have a vector space over $Bbb Z$. By definition, a vector space is required to be over a field. If you take away the field requirement, what you're left with is calles a module. And yes, $Bbb Q^n$ is definitely a $Bbb Z$-module.
That being said, there is a list of $10$ requirements for a space like $Bbb Q^n$ to be a vector space over a field like $Bbb Q$. All of them should be easily verifiable. So yes, $Bbb Q^n$ is a vector space over $Bbb Q$.
answered 40 mins ago
ArthurArthur
113k7110193
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
You can't have a vector space over $Bbb Z$. By definition, a vector space is required to be over a field. If you take away the field requirement, what you're left with is calles a module. And yes, $Bbb Q^n$ is definitely a $Bbb Z$-module.
That being said, there is a list of $10$ requirements for a space like $Bbb Q^n$ to be a vector space over a field like $Bbb Q$. All of them should be easily verifiable. So yes, $Bbb Q^n$ is a vector space over $Bbb Q$.
answered 40 mins ago
ArthurArthur
113k7110193
$endgroup$
add a comment |
$begingroup$
You can't have a vector space over $Bbb Z$. By definition, a vector space is required to be over a field. If you take away the field requirement, what you're left with is calles a module. And yes, $Bbb Q^n$ is definitely a $Bbb Z$-module.
That being said, there is a list of $10$ requirements for a space like $Bbb Q^n$ to be a vector space over a field like $Bbb Q$. All of them should be easily verifiable. So yes, $Bbb Q^n$ is a vector space over $Bbb Q$.
answered 40 mins ago
ArthurArthur
113k7110193
$endgroup$
add a comment |
$begingroup$
You can't have a vector space over $Bbb Z$. By definition, a vector space is required to be over a field. If you take away the field requirement, what you're left with is calles a module. And yes, $Bbb Q^n$ is definitely a $Bbb Z$-module.
That being said, there is a list of $10$ requirements for a space like $Bbb Q^n$ to be a vector space over a field like $Bbb Q$. All of them should be easily verifiable. So yes, $Bbb Q^n$ is a vector space over $Bbb Q$.
answered 40 mins ago
ArthurArthur
113k7110193
$endgroup$
You can't have a vector space over $Bbb Z$. By definition, a vector space is required to be over a field. If you take away the field requirement, what you're left with is calles a module. And yes, $Bbb Q^n$ is definitely a $Bbb Z$-module.
That being said, there is a list of $10$ requirements for a space like $Bbb Q^n$ to be a vector space over a field like $Bbb Q$. All of them should be easily verifiable. So yes, $Bbb Q^n$ is a vector space over $Bbb Q$.
answered 40 mins ago
ArthurArthur
113k7110193
answered 40 mins ago
ArthurArthur
113k7110193
answered 40 mins ago
ArthurArthur
113k7110193
answered 40 mins ago
ArthurArthur
113k7110193
113k7110193
add a comment |
add a comment |
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4
$begingroup$
It doesn't make sense to be a vector space over $mathbb Z$, because $mathbb Z$ is not a field. However, it is a vector space over $mathbb Q$.
$endgroup$
– Ashwin Trisal
41 mins ago
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
39 mins ago