Linear Dependence In \(\mathbb{R}\) Over \(\mathbb{Q}\)

Algebra Level 4

Let \(V\) be a vector space over any field \(F\).

A collection of vectors \(v_1, v_2, \cdots, v_n \in V\) is called dependent if there exist real numbers \(a_1, a_2, \cdots, a_n \in \mathbb{R}\) such that \[a_1 v_1 + \cdots + a_n v_n = 0\] and at least one of the \(a_i\)'s is nonzero. Consequently, a collection of vectors is called independent if it is not dependent.

Note that \(\mathbb{R}\) may be thought of as a vector space over the field \(\mathbb{Q}\) of rational numbers. With this vector space structure on \(\mathbb{R}\), is the set \[\{\sqrt{2}, \sqrt{3}, \sqrt{5}\} \subset \mathbb{R}\] dependent or independent? What about the set \[\left\{\frac{\pi}{4}, \arctan\left(\frac{3}{2} \right), \arctan(2) \right\} \subset \mathbb{R}?\]

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