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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb{R}$ ...
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### 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 ...
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### 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. (...
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### 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 ...
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### 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 ...
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