Suppose V is a real vector space of dimension 3. Then what will be the number of pairs of linearly independent vectors in V? Cani say it should be infinity? Because there exist infinite number of basis of a vector space of dimension 3?
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This depends on the ground field. If the ground field is infinite, this is in fact true and not all that hard to prove (just take two linearly independent vectors $v$ and $w$ and consider the pairs $(v,\lambda w)$ for any scalar $\lambda \neq 0$). If the ground field is finite, the vector space itself is finite as well, so your claim does not hold.
Edit: I missed that you specified it was a real vector space. In that case, the ground field is obviously infinite, and the argument above works. Thanks to Jonathan Taylor for noticing.
Since your question specifies that we are talking about a real vector space, then yes. There are an uncountably infinite number of such pairs.