# 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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