Discussion
Which of the following statements is true ?
Answer: B
The true statement is that R is a vector space over Q of dimension \(\infty\).
Q is countable, so every finite-dimensional vector space over Q is countable. R is uncountable, so no finite basis can exist and the dimension is infinite (in fact uncountable).
The others are false. C over R has basis {1, i} and dimension 2. Polynomials of degree at most n have basis \(1, x, \ldots, x^{n}\), which is \(n + 1\) vectors, not n. And only a linearly independent set can be extended to a basis; a dependent set such as {v, 2v} cannot be.
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