Linked Questions

5
votes
3answers
3k views

Prove that this vector space is not finite dimensional. [duplicate]

Let $V$ be the set of real numbers. Regard V as a vector space over the field of rational numbers $F$ with the usual operations. Prove that this vector space is not finite dimensional. My attempt: Let ...
9
votes
2answers
2k views

Dimension of R over Q without cardinality argument. [duplicate]

I am looking for the easiest (elementary) proof that $\mathbb R$ is infinite dimensional as a $\mathbb Q$-vector space, without using cardinality. It should be understandable at highschool level. So ...
0
votes
1answer
4k views

Dimension of R over Q [duplicate]

Possible Duplicate: Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? consider R as a vector space over Q then what is the dimension and ...
2
votes
2answers
138 views

Infinite dimensional vector space [duplicate]

I am solving the following : Vector space $\mathbb{R}$ over rational number $\mathbb{Q}$ is infinite dimensional. I proved this by using that $\mathbb{R}$ is uncountable, but my professor ...
0
votes
0answers
118 views

Real numbers as a $\mathbb{Q}$-linear space [duplicate]

Let's consider the real numbers as a $\mathbb{Q}$-linear space and denote it by $\mathbb{R}_{\mathbb{Q}}$. It was proved here that $\dim \mathbb{R}_{\mathbb{Q}} > \aleph_0$. I suspect that the ...
0
votes
0answers
17 views

what is the dimension of $\mathbb{R}$ at a vector space over there field $\mathbb{Q}$? [duplicate]

If we look at $\mathbb{C}$ as a vector space over $\mathbb{R}$ it's dimension will be $2$, because $\mathbb{C} = span\{1,i\}$. A question I thought of is what would be the dimension of $\mathbb{R}$ ...
127
votes
3answers
15k views

The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using elementary ...
20
votes
13answers
4k views

What are examples of vectors that are not usually called vectors?

In algebra, a vector is an element of a vector space. An example of such an element is a matrix. In linear algebra, a vector is a shorthand name for a $1 \times m$ or a $ n \times 1 $ matrix. (...
41
votes
6answers
55k views

Proving Irrationality

How is it possible to prove a number is irrational? First part of that question: How it possible to know that a number will go on infinitely? Second part: How is it possible to know that no ...
8
votes
4answers
4k views

Why do we call a vector space in terms of vector space over some field?

I am getting a bit confused with the terminology here. I understand that a field means some set of scalars like real numbers but why do we need a field for a vector space? Are not the numerical values ...
8
votes
4answers
2k views

Isomorphism of Vector spaces over $\mathbb{Q}$

From this post we see that $\mathbb{R}$ over $\mathbb{Q}$ is infinite dimensional. Similarly $\mathbb{C}$ over $\mathbb{Q}$ is also infinite dimensional, and I rememeber having solved a problem that $\...
5
votes
2answers
476 views

A question about a method that shows $\mathbb{R} $ not finite dimensional.

Upon looking at methods that show $\mathbb{R}$ is not finite dimensional over $\mathbb{Q}$ I came across a method mentioned here by the user Bill Dubuque, he took a set of vectors of the form $\log(p)$...
2
votes
3answers
109 views

If $x^3$ is a square, is $x$ a square?

Very simple question here, which I feel like I should be able to answer but am struggling with. Let $k$ be a finite field, and let $x\in k^\times$. Is it true that $$x^3\in\left(k^\times\right)^2 \...
0
votes
4answers
785 views

What does “Consider R as an vector space over Q” mean?

As a bonus question in an exam, I had to prove that R is infinite dimensional as a vector space over Q. I would have probably tried cardinality to show it, but I don't know what it means..
4
votes
1answer
789 views

Analogy between linear basis and prime factoring

I recall learning that we can define linear systems such that any vector in the system can be represented as a weighted sum of basis vectors, as long as we have 'suitable' definitions for addition and ...

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